Omnipotence has never been adequately defined, either in the Bible or by philosophers or theologians. It doesn’t mean there are no limits to what God can do. It does mean that God is able to do all things that are proper objects of his power. It is not a contradiction that God can make happen whatever is possible, but that no number of actualized possibilities exhausts his power. God can realize whatever is possible. The Bible talks about the omnipotence of God as all-sufficient power. He is able to overcome apparently insurmountable problems. He has complete power over nature. He has power over the course of history. He has the power to change human personality as individuals allow. He has the power to conquer death and sin, and to save a human soul for eternity. He has power over the spiritual realm.
What all of this means is that God’s will is never frustrated. What he chooses to do, he accomplishes, for he has the ability to do it. But that's not as complete a definition of omnipotence as we might hope for.
There are, however, certain qualifications of this all-powerful character of God. He cannot arbitrarily do anything whatsoever that we may conceive of.
- He can’t do what is logically absurd or contradictory
- He can’t act contrary to his nature
- He cannot fail to do what he has promised
- The theology of omnipotence rejects the possibility of dualism
- He cannot interfere with the freedom of man
- He cannot change the past
- It is not violated by self-limitation on the part of God
- It does not imply the use of all the power of God

Another aspect of God’s omnipotence is that he is free. Nothing in Scripture suggests that God’s will is determined or bound by any external factors. God’s decisions and actions are not determined by consideration of any factors outside himself, but are simply a matter of his own free choice. He has the power to do what he chooses to do and bring about what is possible. But that's not a good definition of omnipotence either. The problem is that there IS no good definition of omnipotence.
Leibniz & Ross philosophically state omnipotence in what’s called a “result” theory: theories that analyze omnipotence in terms of the results an omnipotent being would be able to bring about. These results are usually thought of as states of affairs or possible worlds: a way the world could be. A possible world is a maximal consistent state of affairs, a complete way the world could be. The simplest way to state it may be, “for any comprehensive way the world could be, an omnipotent being could bring it about that the world was that way.” Ross formulated it as “Since every state of affairs must either obtain or not, and since two contradictory states of affairs cannot both obtain, an omnipotent being would have to will some maximal consistent set of contingent states of affairs, that is, some one possible world.”

I'm not particularly familiar with the Bible, but does it actually state that god can do literally anything? Certainly it states that he created the universe, created man and all the animals, etc. But is it actually spelled out that there is literally nothing beyond god's power?

If not, where'd the idea of an omnipotent deity come from?