There is an argument by a philosopher named Roland Puccetti showing, convincingly, I think, that omniscience (i.e. all-knowingness) is impossible. It is wonderfully called the "argument from negative universal existential statements." I have read his argument in Michael Martin's book Atheism: A philosophical justification and on the internet. However, I think I can more clearly explain it here, and I am sharing it with others because it is often useful to actually go into attack mode when debating believers in an omniscient God. The argument first assumes one can know everything and then shows by where it leads, that this is not possible.
If you knew everything, your knowing everything would be a fact.
If you knew everything, you would also have to know the fact that you knew everything.
Facts are "infinite" in quantity, because mathematical facts are infinite, and they are a proper subset of all facts.
However, you could never know the fact that you know everything, because in being infinite in quantity, you can never know all facts. However many you know, there are still infinitely more to know.
Since you can never know the fact that you know everything, you cannot be omniscient. (Neither can God.)
Objection 1: Mathematical facts do not really exist.
Reply: Even if mathematical facts oddly enough don't really exist, distances, motions, and durations do exist, and there are an infinite number of distances between particles moving in relation to each other over a period of time.
Comment: It is interesting to think about the fact that there is an infinite "number" of facts, yet there are only a finite number of ways to store facts as usable information, because particles in the universe are finite. It also shows that trade-offs have to exist. You may in certain circumstances be able to know one thing or the other, but never both. I have had two physics classes, two physics labs, and a lot of chemistry so far, and I think the Heisenberg Uncertainty Principle is one convincing example of this. I also think there are likely others. If anyone knows any, please post them!
Mathematical facts aside, if god knew everything there would be no point to the entire bible. The very idea of salvation is absurd. People have faith and they are so worried about it when its only meaning (salvation) is that god is playing a game with you. Tell this god that you do not want to play the game.
There is no contest when the winner is known in advance.
Also, if "God" knew everything, thinking would be redundant for such a being. Thinking is how we arrive at new information.
Commenting upon my own post, it occurred to me that if you have an irrational number, such as the square root of two, there is a limit as to how many digits of it can be expressed or stored in all the matter of the universe. If "X" is the last digit of the square root of two that can be expressed or stored starting from the first digit, on account of the limited matter available in the universe, you can either know the square root of two from the beginning up to the Xth digit or the digits of the square root of two from the Xth digit to the 2Xth digit, but not both. This is an example of the type of trade-off at which I insinuated above.