[1]  In the phrase "the power to bring about any state of affairs", the key word is any- thereby including impossible states (such as a spherical cube or 2 + 2 = 5). I am a bit skeptical that we as humans yet fully know what is impossible and what is not; (a century ago no one would have believed many of the technological advances that we now take for granted). But common sense, at least as it stands presently, doesn't allow for spherical cubes or math that doesn't follow its own rules. And therefore the idea of an omnipotent being that can create impossibilities is incoherent.

     A God that has maximal power, in contrast, still meets the requirements of omnipotence without the paradoxes of impossible states of affairs. Such an entity would have more influence, opportunity, and ability to effect change than any other being. This second reckoning is the more plausible, as It conforms to the logic we have observed in the universe. But still- I don't think anyone can know, with absolute certainty, whether or not a God could perform impossibilities. Infinity, for example- the concept of time never ending seems an impossibility. Yet mathematics says otherwise.

     Two omnipotent beings? Not logical, because they would cancel each other out- if God-1 wants a rose to be red, and God-2 wants it to be blue, and each God has maximal power, then what is going to happen? (I'm only half-joking with a prediction that the rose would be purple). But since that isn't what either God wanted, the result is that neither is omnipotent. It has some interesting implications. A maximally powerful being could not clone itself, apparently. More puzzling to me is how to quantify maximal power. Just as numbers and time never end, it would seem that power could always increase, never reaching a limit.

[2]  The paradox of the stone is a convenient thought-experiment for analyzing omnipotence- for an all-powerful being to create something too heavy for He himself to lift is a seeming impossibility. And if one considers the premise that a God should be able to bring about any state of affairs, then this is at least one state that cannot be brought about.

     One solution that H&R offer is for such an omnipotent being to have the ability to temporarily lose that omnipotence; creating a sufficiently heavy stone (while omnipotent) so that it cannot be moved once that power is lessened. It's a logical way around the problem. Except that one has to wonder: if a God temporarily abdicates maximal power, how can He get it back?

[3]  "Contingent states of affairs which an omnipotent being is powerless to bring about under certain circumstances." Contingent as an adjective has two meanings- 1. "subject to chance", and 2. "dependent on." Both definitions have some bearing on these events.

(1) that a raindrop fell 
(2) that a raindrop fell at t (where t is a past time)

     Both statements refer to a past event, with the latter more specific as to which moment in the past. Even an omnipotent being cannot change what has already occurred, nor even be the cause of a past event from the present- although we can assume this entity did cause the raindrop to fall when it happened.

(3) that Parmenides lectures for the first time   

     There's no significant difference here from the raindrop falling, except that Parmenides as a human perhaps had free will as to when he would lecture. After the fact, neither an omnipotent being nor Parmenides can change it.

(4) that the Amazon river floods an odd number of times less than four

     This one I found odd (pun intended). The two options numerically are 1 and 3, obviously. After the first flooding, an omnipotent being could indeed either keep the Amazon from flooding again, or make sure that it floods two more times. Either way the entity does bring about these events. But again, once they've happened, no entity can change that. (And we have to assume that the Amazon has flooded many more than just three times).

(5) that Plato freely decides to write a dialogue          

     For Plato to freely decide implies free will, and that an omnipotent being had no influence on the event- before, during, or after.

     While these thought-experiments help define what omnipotence is (and is not), I don't find a great amount of significance in the fact that a God cannot change the past. (The universe would be a very chaotic place if one could). In looking around the internet, I found a quote by Epicurus, the ancient Greek philosopher:

"Is God willing to prevent evil, but not able? Then He is not omnipotent. Is He able, but not willing? Then He is malevolent. Is He both able and willing? Then whence cometh evil? Is He neither able nor willing? Then why call Him God?"

     I know we'll get into that more in a later reading... but to me these are the most relevant questions to be asking about an omnipotent being.

@Amy Olsen:

     As always, you do a thorough and insightful job on this topic. Omnipotence has a lot of gray areas, in my opinion. While I understand philosophy's goals (at least to a certain degree), I have to question how productive it actually is to ponder things like "stones too heavy for God to move." I understand that philosophy, by its nature, is somewhat elitist. The "discipline" never existed (at least formally) until humans had advanced enough for some to have "leisure time", and actually be able to just sit around and think, as opposed to ancient cultures where every waking hour was purely about survival.

     A large percentage of humans believe in a God; a fairly large amount does not. Neither side, as yet, has been able to prove their perspective. And short of God making His presence known in some undeniable way (like, actually becoming visible and audible to us), I doubt if the debate will ever end.

    One has to wonder- why has there never been a single, verifiable, instance of communication from God? Why play games and make us have "faith?" How many of us would willingly accept the same terms from another human- "just do what I say, or suffer the consequences?"... at least we would know that the human making those demands exists.

    Some people believe in extraterrestrials; some in various cryptozooligical entities (Bigfoot, Loch Ness monster, etc); some in angels... and some in God. What do they all have in common? Not one single shred of evidence to date...

    * It should go without saying, but this is purely a rebuttal to certain concepts- and not you personally, Amy... I both admire and envy your faith. Your open-mindedness about it is refreshing and you never fail to make me re-think my own doubts...

@William Bryan:  

    The third question in this discussion, with the examples of past events, is the most intriguing for me. I'll just focus on one, "a raindrop fell." Assuming that there is an omnipotent being, it's safe to say that this being caused the raindrop to fall, in the moment that it happened. It's possible that this "God" even knew well beforehand that He was going to make the raindrop fall (a fairly trivial thing, for sure, but if one is a God, omnipotence and omniscience are part of the package. "It's what you do").

     So, God had influence on the raindrop in the past before it fell, and also in the precise moment that it fell. I've been trying to find a way He can still influence that raindrop now that it has presumably long since hit the ground.

     And in writing that last sentence, I might have stumbled on a solution. Maybe God could have decided that this particular raindrop was special. God decided that, unlike other raindrops, this one would fall extremely slowly- like one centimeter every million years. It stays intact (doesn't evaporate) and is still falling, right now and for a very long time to come. And God knows this.

     It's a stretch, I know... but true omnipotence and omniscience would imply that God could do such things, and would also know where all the molecules of any normal raindrop have traveled since it fell. If complete knowledge of all the repercussions of a given event are retained by God, then in a sense He still has an influence over past ones- He knows all the consequences as they continue to unfold.