If you ask Rick Astley for his copy of Disney Pixar's Up, he can't give it to you, because he'll never give you Up. But by not doing so, you'd be let down, and he'll never let you down.
Testing this scenario is ofc incredibly risky to the state of our reality, so the Astley paradox must remain a thought experiment.
I guess I would say the paradox of tolerance. I'm sorry but I'm just gonna yoink the definition from Wikipedia because I'm not great at explaining things:
The paradox of tolerance states that if a society's practice of tolerance is inclusive of the intolerant, intolerance will ultimately dominate, eliminating the tolerant and the practice of tolerance with them. Karl Popper describes the paradox as arising from the fact that, in order to maintain a tolerant society, the society must retain the right to be intolerant of intolerance.
Bonus least favorite paradox: You need experience to get a job and you need a job to get experience.
The Unexpected Hanging Paradox: A man is sentenced to death, but the judge decides to have a little fun with it. The man will be killed at noon on a day of the judge's choosing in the next week, from Monday to Friday. The only stipulation is that the man will not expect it when he's called to be killed.
The man does some quick logic in his head. If Friday is the last day he could be killed, then if he makes it to Friday without dying, he knows he must die on that day. And since that wouldn't be a surprise, he cannot be killed on Friday.
He then extends the logic. Since he can't be killed on Friday, the last day he can be killed is on Thursday. Thus, all the prior logic regarding Friday applies, and he cannot be killed on Thursday either. This then extends to Wednesday, then Tuesday, and then Monday. At the end, he grins with the knowledge that, through logic, he knows he cannot be killed on any of the days, and will therefore not be killed.
Therefore, the man is astonished when he's called to be killed on Wednesday.
Mine is similar to yours in that it's about the power of God. It's called the Epicurean Trilemma:
If a god is omniscient and omnipotent, then they have knowledge of all evil and have the power to put an end to it. But if they do not end it, they are not omnibenevolent.
If a god is omnipotent and omnibenevolent, then they have the power to extinguish evil and want to extinguish it. But if they do not do it, their knowledge of evil is limited, so they are not omniscient.
If a god is omniscient and omnibenevolent, then they know of all the evil that exists and wants to change it. But if they do not, which must be because they are not capable of changing it, so they are not omnipotent.
This proves fairly simply that God as commonly interpreted by modern Christians cannot exist. Early Christians and Jews had no problem here, because their god was simply not meant to be omnibenevolent. Go even further back in time and he was not omnipotent, and possibly not omniscient, either. "Thou shalt have no gods before me" comes from a time when proto-Jews were henotheists, people who believed in the existence of multiple deities while only worshipping a single one.
My favorite paradox is the "Stay signed in" option Microsoft gives you when signing in. Because despite keeping you signed in on every other site in existence, Microsoft, who is usually hooked into your OS, does not. Thus, stay signed in runs contradictory to one's expectations.
Not sure if its what you're talking about but I really like the Ship of Theseus thought experiment, if an object is the same object after having had all of its original components replaced. Always makes me think of if an exact clone of you is created (same thoughts, memories, etc...) should that be considered you?
Zeno's Paradox, even though it's pretty much resolved. If you fire an arrow at an apple, before it can get all the way there, it must get halfway there. But before it can get halfway there, it's gotta get a quarter of the way there. But before it can get a fourth of the way, it's gotta get an eighth... etc, etc. The arrow never runs out of new subdivisions it must cross. Therefore motion is actually impossible QED lol.
Obviously motion is possible, but it's neat to see what ways people intuitively try to counter this, because it's not super obvious. The tortoise race one is better but seemed more tedious to try and get across.
There's so many good ones, and I'd probably say Russel's (what's in the set of every set that doesn't contain itself?), but recently the unexpected hanging has come up a couple times. That one is all about how theories or rules can break if they become contingent on how an observer is thinking about them (including state of knowledge of the situation).
God clearly can't exist because an omnipotent, omniscient, and just God is a paradox already. Omnipotence and omniscience means that God, if they exist, would have full control of every moment of the universe (even if they only "acted" initially). Some (I'd argue nearly all) people suffer for reasons out of their control. Only deserved suffering is just. Since undeserved suffering exists then God cannot exist (at least omniscient, omnipotent, and just - as we understand those terms). God could be an omniscient, omnipotent asshole or sadist... God could be omniscient and just (aka the martyr God who knows of all suffering but is powerless to prevent it)... or God could be omnipotent and just (aka the naive God who you could liken to a developer running around desperately trying to spot patch problems and just making things worse).
Alternatively, by omnipotent maybe the scriptures are just hyping them up - "God is so fucking buff - this one time they lifted up this rock that was like this big. Fucking amazing."
Fermi's Paradox. There are so many stars (more than there are grains of sand on earth), that the probablility that one of them has life, and even intelligent life, is >99% . So why haven't we observed it yet? Cue a lot of brilliant people trying to answer that question.
In gridiron football, if a penalty is committed close enough to the end zone, instead of the normal penalty yardage, the ball is spotted half the distance to the goal (i.e. if a defender holds an offensive player and the offense is 8 yards away from the end zone, instead of being penalized the normal 10 yards they would be penalized 4). In theory, there can be an infinite amount of penalties to the point where penalties would move the ball micrometers or even shorter without the ball ever crossing the end zone.
There's probably a name for this phenomenon, but I can't think of it.
Basically, a super-powered future A.I. that knows whether or not you will build it. If you decide to do nothing, once it gets built, it will torture your consciousness forever (bringing you "back from the dead" or whatever is closest to that for virtual consciousness ability). If you drop everything and start building it now, you're safe.
Assuming time travel exists: is it possible to alter the past?
If an event occurs, and you decide to travel back in time to change/prevent that event: It has no longer occurred in the way that caused you to want to change it; thus you never travel back to change it, and it does occur...
Newcomb’s paradox is my favourite. You have two boxes in front of you. Box B contains $1000. You can either pick box A only, or both boxes A and B. Sounds simple, right? No matter what's in box A, picking both will always net you $1000 more, so why would anyone pick only box A?
The twist is that there's a predictor in play. If the predictor predicted that you would pick only box A, it will have put $1,000,000 in box A. If it predicted that you would pick both, it will have left box A empty. You don't know how the predictor works, but you know that so far it has been 100% accurate with everyone else who took the test before you.
If you have a sword that can cut through anything, and a shield that can absorb any damage unharmed, what happens if you swing the sword at the shield?
In classical logic, trichotomy on the reals (any given numbers is either >0, <0 or =0) is provably true; in intuitionistic logic it is probably false. Thanks to Godel's incompleteness theorem, we'll never know which is right!
How can you kill a god? What a grand and intoxicating innocence! How could you be so naive? There is no escape. No Recall or Intervention can work in this place. Come. Lay down your weapons. It is not too late for my mercy!
A similar one would be can God create a rock so heavy he can't lift it. The problem with these statements is that they're not logically sound. As this would be akin to saying, can god be god and not be god at the same time? Which is contradiction and syntactical jargon. A simpler example is like someone saying they have a squared circle.
Why is "can god kill god" a paradox? They either can or they can't (picking "they" because your particular god might not be a he). If they're all-powerful then the answer is yes, because they can do anything. I don't see how that's paradoxical.