The lesson I wished to draw from the example of adverse possession is the importance of possession and use to establish claims to property rights. I did not intend to make the stronger claim that there is an analogy between adverse possession and the eventual ceding of intellectual property rights, and in fact there is not an analogy to be made. So please, lets back up a bit.
Okay, we're agreed that there's not an analogy to be made. (Though I thought you were drawing such an analogy when you'd said "Every user of a patented invention or owner of a book is as a squatter on a piece of real estate." I think you'll understand my confusion.)
There is some ambiguity in the use of the word 'rights'. It is a bridging concept between the field of ethics and the field of politics, and it is also used to refer to privileges constructed by contractual agreement (i.e. 'a right to exercise an option'). It can sometimes be necessary to clarify which context applies. In the case of adverse possession, there was a mismatch between what the legal formalities of title to a land specified and the facts on the ground. Ethically, the formal owner had no rightful claim on the land due to abandonment and so then politically (judicially) the legal title is amended to be correct.
I'm new to the "adverse possession" thing -- you may well be right. Though it's unclear to me that this is strictly an issue of "abandonment" rather than someone new seizing control. After all, it does not seem to be at issue how long it's been since the owner last tended to his land, but rather how long the new claimant has held continuous possession of it. (If an owner had clearly abandoned the land in question, but the new claimant was himself new to the scene, I don't know that he would take legal possession. Or there might be other legal means of determining "abandonment," but I'm just addressing myself to "adverse possession," as represented in that Wiki.)
I think that we're agreed that law of this nature -- dealing with issues of property abandonment -- is important, however, if that speaks to your point in raising it. Though a man owned a house, if he's been dead and buried for a hundred years, and the house abandoned all the while, and if someone else sets up there and makes a life for themselves, that is their property and needs to be recognized as such... even if thereafter descendants of the original owner come a-knocking.
In the absence of any legislated IP law inventors and authors would attempt to protect their interests via uses and misuses of contract law, the contracts being conditions imposed upon the sales of the inventions and texts and artwork. The current proliferation of end user license agreements (EULAs) for software is an example of an attempt to defend IP with contract law. To repeat myself, in the absence of legislated IP law we would instead have a judicial case law for IP entirely within the specialty of contract law, judging the validity and permissible scope of the clauses restricting reproduction and resale. What would you have to say about that, would you want to ban those contracts?
I have no interest in abridging or eliminating any area of the right to contract (though we're agreed that certain kinds of contracts are impermissible), nor do I understand why you would think I would. If two people wish to enter into any specific agreement, far be it from me to intervene. If you would say that the ends at stake are the same, that may well be, but I'm also interested in ensuring that the means are appropriate. Here, I think a person should recognize those rights of others which can be objectively demonstrated to exist (which includes property, but not "intellectual property"), and also be beholden to those contracts which he has personally agreed to. I would not hold him to "contracts" to which he has not given consent.
And come to it, I'm not certain that contract law, either in theory or in practice, would take the place of intellectual property, point for point. For instance, consider this case which Ayn Rand addressed in her essay on Patents and Copyrights:
As an objection to the patent laws, some people cite the fact that two inventors may work independently for years on the same invention, but one will beat the other to the patent office by an hour or a day and will acquire an exclusive monopoly, while the loser’s work will then be totally wasted. This type of objection is based on the error of equating the potential with the actual. The fact that a man might have been first, does not alter the fact that he wasn’t. Since the issue is one of commercial rights, the loser in a case of that kind has to accept the fact that in seeking to trade with others he must face the possibility of a competitor winning the race, which is true of all types of competition.
If we deal with these matters according to the contracts people willingly enter into (i.e. according to their actual rights), then the answer to this question becomes very different.
As the source of and first possessor of a new article of intellectual property it is only just that the artist or inventor be permitted to commercialize his property and sell it on any terms he pleases including restricting resales or reproductions, i.e. exercise his right of property.
I would say that anyone who owns a piece of property is permitted to commercialize his property and sell it on any terms he pleases (subject to those things we agree are not permitted generally). I don't agree if you would hold to the seeming implication that "the source and first possessor" has new/different rights, above and beyond anyone else. I believe that everyone has the same property rights generally, though different specific property.
But he cannot exercise those terms without limit because his customers have property rights as well, right founded on actual use and possession in the present rather than the remote past, up to twenty years in the past for inventions (and much longer for copyrights as they are not utilitarian and therefore less likely to be some second person's means of livelihood).
Well, I don't know if I agree that a specific, contractual limit to reproduction can not last longer than twenty years. If a person offers an "invention" with disagreeable contractual limits on reproduction, a person can always opt not to accept the deal. And as for this division between "utilitarian" patents and not-as-such copyrights...?
It's important to me, from time to time, to reintroduce the wider context in which we're working. In discussing intellectual property, we're supposedly dealing with man's rights -- the violation of which is an initiation of the use of force. Unless that kind of thing is just accounted empty rhetoric? But I've never taken it that way -- I take it seriously and literally. So if "twenty years" is the wrong figure, one year too long or two short in any individual case, then we are necessarily on the side of the initiation of the use of force in that case.
So to entrust such things to what we deem "less likely to be some second person's means of livelihood"...? Strikes me as
way wrong. If what's at stake here is a person's means of livelihood, and their rights (which I relate back to being a life-or-death kind of thing), then I certainly don't want a one-sized-fits-all adjudication according to what "seems likely."
The boundary between the rights of the sellers and the rights of the buyers could be left to each product's contract of sale and each court case in each separate jurisdiction or we could have a uniform code of law for all IP products in all jurisdictions.
I prefer the uniformity and believe that best defends everyone's rights and enhances the trade value for everyone concerned. You may be unconvinced of the advantages of uniformity, but don't believe for a second that repealing IP laws would prevent attempts to control IP by other non-uniform means.
It's not that I'm unconvinced "of the advantages of uniformity," but that I don't like the cost, if that cost involves curtailing man's rights. There are all sorts of things that are non-uniform in society, because they're left to individual contracts and cases, including wages, prices, and etc. It's messy. But I'm pretty sure I'd be arguing against any proposal to give uniformity to any of those things, despite whatever apparent advantages such uniformity might arguably confer (and frankly, I suspect you and I would be in full agreement in those cases).
You would be cutting the head off of a hydra, which would grow ten-thousand heads and more to replace the one lopped off. If your ambition is to kill the IP hydra, you could not do it except by assaulting the freedom of contract in general.
I like the imagery!

But no, I've no ambition to "kill the IP hydra." I've started to believe that IP is a paper creature, whatever its species -- a phantasm. And yes, I do want to banish any such shadows with... er... the candle of truth (I should really leave the imagery to you), but not for the sake of remaking and controlling society in some specific fashion; I want man's rights to be clearly recognized that they might be clearly defended. This is important to me. Whatever results would be duplicated through a rightful use of contract law, are fine with me. Whatever would be gained would be gained, whatever would be "lost" would be lost. I fight IP, not because I hate patents, per se, but because I hate that which is false. Yet I think the results would be better all around, because, in principle, I believe that rights -- proper rights, properly understood -- work like that.
Addendum to my last post: The Doctrine of Right of First Sale has its basis in copyright law. Without copyright law, there would be no Doctrine of Right of First Sale and you might today be not able to sell your used DVDs, games and other stuff.
On the subject of selling used games, and etc., I can't rightly predict what would happen "in a world without IP." If someone offered to sell me a game with the stipulation that I would not re-sell it for X amount of time, and if I agreed to that stipulation in my purchase, then I guess I should be held to it. Is this an unpalatable outcome? (This assumes, of course, that such contract provisions were held as proper, generally speaking... though I suspect that if you think the Doctrine of Right of First Sale is appropriate as a limitation to IP, you might hold it appropriate as a limitation on these kinds of contracts, too.)
a. I hope your childhood traumas are not so serious that they impair your ability to reason.
I think I'll choose not to read this in a too-insulting fashion. I'd like to preserve the bonhomie of a good back-and-forth for as long as reasonable.
But no, I don't expect that my "traumas" (which can also be called "memories," if we're feeling generous) impair my ability to reason, given that I remain on the correct side of this issue. Although... were my ability to reason sufficiently impaired, I guess I wouldn't know it!

Ultimately, you'll have to weigh my arguments, such as they are, and decide for yourself.
As a further FYI, I share such information (apparently at my peril) partly in an effort at a more-honest communication. It isn't the case that at some early age I explicitly identified certain axioms, and I've just been working deductively from there on out. Sometimes my actual experiences weigh in on my thoughts and feelings. And when I recognize that -- when I have some insight into my own process -- I try to share it. Hopefully, with this context considered, the memories and experiences that I choose to share here can become more than fodder for an implication that perhaps the reason we disagree is that I'm psychologically impaired.
b. Am I scary like that? 
I don't know that I consider my experience to be "scary," from my current perspective, or that there was much fear communicated in what I'd written. At the time it was unpleasant, that's for sure, though I never sought to make any amends for what "I'd done" -- I just kept on reading my book. For better or worse, my strategy has most often been to "ignore the haters" (though I do make occasional exceptions). In discussing IP, which my anecdote might be understood to cast as "law as enacted by petulant adolescents," I guess the "haters" have guns at their disposal, to force me to act against my benefit (and after all, perhaps John did have some sort of special idea claim on "the reading of Jurassic Park"? If I adopted the mentality and arguments that give support to IP, I think I'd at least have a shot at making his case).
But to answer you directly, no, you're not scary. I'm not "afraid" of intellectual property, except that I think it's a violation of man's rights... which is a pretty scary thing, I guess. So... maybe?

(What
is scary to me, though I try not to give it much thought, is the question of... if I'm right about this whole IP deal -- as I think I am -- what does it mean that so many prominent Objectivists are apparently so convinced of the legitimacy of IP? But that question won't help me resolve the actual matter, so I'm leaving it for later consideration.)
Because justice is not automatic. Justice is man-made and if no one acts to achieve it then it will not exist.
Perhaps. I think that the simpler reason as to why the innovator in my example is not wealthy with a bazillion dollars, is that he has not done what is necessary to be wealthy in fact -- he has not created the wealth. He has not created those items which people would trade their bazillion dollars to acquire. Those items do not yet exist. What exists is an idea. But that idea does
nothing for people until it is put into practical effect, and if the innovator cannot do that -- if he does not in fact do it -- then he will create nothing.
I think that the person who creates the wealth (as in: actually creates the wealth, so that there exists more than nothing) will be the one who winds up wealthy. And this makes a rough sort of sense to me, in the same way that property belonging to he who invests the effort to create it makes sense.
If I'm right, then the "justice" you're talking about is a redistribution of wealth, from those who've created it to those who have not... which is to say, not justice at all.