If an algo
X outputs the minimum "number of moves" (not the actual moves, just their count) to solve an input configuration, what's the maximum number
X can output for any configuration?
If an algo
The maximum number of positions a valid Rubik's cube configuration takes to solve is 20, and is called God's Number. No rigorous mathematical proof has been provided for the number; however, every configuration was tried. Cube20 gives an explanation of what they did, as well as provides source code for their project.
They achieved this through a few groups, discovered from other proofs of maximum move counts, which reduced the set of positions that actually needed to be tried significantly. They only needed to solve 55,882,296 positions as a result (they took quite a bit of advantage of symmetry).
However, without brute-force, this has only been mathematically proven for 22 moves.
It depends on what you count as a move. The "God's number is 20" result mentioned by earlier answerers is for the face turn metric, where a 180 rotation/half turn of a face counts as one move. (This blog entry by David Joyner gives a history of results toward this conclusion with a parallel human interest story on one of the primary researchers.)
In the quarter turn metric, a 180 rotation counts as two moves so that "God's number" is presumably higher. To date it is known to be at least 26 ("superflip plus fourspot", description here). Similar to the long-known examples requiring 20 turns in the FTM that were eventually shown (with lots of computer power) to be optimal, the suspicion is that 26 is God's number in the QTM.
I think that's the question you were asking. If not, please edit your question to clarify.