The fundamental limitations are closely related to those first identified by Gödel [10]: Any formal system that encompasses arithmetics is either flawed or allows for unprovable but true statements. Hence even a Gödel machine with unlimited computational resources must ignore those self-improvements whose effectiveness it cannot prove, e.g., for lack of sufficiently powerful axioms. In particular, one can construct examples of environments and utility functions that make it impossible for the Gödel machine to ever prove a target theorem. Compare Blum's speed-up theorem [3,4] based on certain incomputable predicates.
Similarly, a realistic Gödel machine with limited resources cannot profit from self-improvements whose usefulness it cannot prove within its time and space constraints.