No, because GPT-4 has finite memory, its context length, and its random number generator for output selection is probably pseudo-random with finite memory.
If the random number generator is pseudo-random, this makes GPT-4 a deterministic finite-state machine, and the output sequence does not necessarily contain all possible subsequences no matter how many times the monkey types a new random key. Put differently, some output subsequences may be inaccessible no matter which keys are input. Same if the random number generator is truly random but its value cannot select among all possible output tokens, only a subset provided by the GPT at each step.
If the random number generator is pseudo-random, this makes GPT-4 a deterministic finite-state machine, and the output sequence does not necessarily contain all possible subsequences no matter how many times the monkey types a new random key. Put differently, some output subsequences may be inaccessible no matter which keys are input. Same if the random number generator is truly random but its value cannot select among all possible output tokens, only a subset provided by the GPT at each step.