We consider the following problem for various infinite-time machines. If a real is computable relative to a large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? We show that the answer is independent of $\mathrm{ZFC}$ for ordinal Turing machines with and without ordinal parameters and give a positive answer for most other machines. For instance, we consider infinite-time Turing machines, unresetting and resetting infinite-time register machines, and $\alpha$-Turing machines for countable admissible ordinals $\alpha$.