Since this was resurrected, here is the statement that at this time seems to me to have the greatest gap between obviousness of truth and obviousness of proof:

• There exists a natural set theoretic universe in which every subset of [0,1] is Lebesgue measurable, so that the reals admit no well order and do not inject into aleph-1.

Here are a different class of obvious theorems, these are only obvious in the sense of physical intuition. They took a long time to prove:

• The existence of solid matter occupying space (in the lowest energy state, the electron-nucleus system occupies a volume proportional to the number of nuclei)
• The positive energy theorem--- every asymptotically flat solution of GR obeying the appropriate energy condition has a positive mass at infinity, with zero mass only for Minkowski space.
• Hard sphere collisions on a negatively curved space are ergodic.

Here is a physically obviously true statement, which can be seen from physical intuition, but which is not proven (as far as I know):

• The asymptotic fate of GR with a positive cosmological constant is within any causal patch, and except for a set of initial conditions of measure zero, a deSitter space.

The reason this is obvious is because the deSitter space maximizes the horizon area, which is a measure of the entropy.