A few things can happen at that point:
First, we can realize that science have never once proven any one of its theories to be mathematically true, is has just bounded the uncertainty to smaller and smaller amounts until we've become convinced that it's close enough to "true" to simply call it that. So once we "prove" that there's no more depth to explore, we may simply have to go back and try to test our theories about the universe at a shallower depth. We've never proven energy is conserved, we've just never found any reason to question it to date (other than the pesky big bang, where we continue to refine our hypotheses to better fit our understood conservation laws).
Beyond that, we should be questioning concept like "what are we?" What am "I?" Books like Douglas Hofstadter's I am a Strange Loop explore this question. What does it mean to "know something?" It turns out that, for mathematical reasons, the concept of "I" is really challenging for science to approach and try to define, partially because the typical meaning of "I know something" involves "I," making it hard to objectively define "know."
From there, pick your branch of philosophy, and the answers will vary. We might become Nietzche's "Last Man," battling for immortality. We may decide to give up our knowledge and truth, and just seek release from suffering, as appears in some religions. The concept of "I" may vanish all together, under the onslaught of its own knowledge, or we may become "one with the Dao."
Then there's some who suggest that this may never even happen. Dan Willard constructs some fascinating mathematical worlds in his research which defy conventional explanation, such a self-verifying systems of axioms where a provably countable infinity outside the system can be provably uncountable inside the system, raising all sorts of questions about the nature of the mathematics we use to underpin our science. Even mathematics, the bedrock of science, is constantly in flux, growing and shaping itself as new information arises.
So keep digging! At the moment, there is no scientific proof that there even exists a limit to human ingenuity. There's theories, no doubt, but no proof. Keep searching!