What would be ideal would be a way of establishing the minimal required exploration rate.
Do you mean a way of establishing this independent of the prior, i.e., the agent will explore at some minimum rate regardless of what prior we give it? I don't think that can be right, since the correct amount of exploration must depend on the prior. (By giving AIXI a different bad prior, we can make it explore too much instead of too little.) For example suppose there are physics theories P1 and P2 that are compatible with all observations so far, and an experiment is proposed to distinguish between them, but the experiment will destroy the universe if P1 is true. Whether or not we should do this experiment must depend on what the correct prior is, right? On the other hand, if we had the correct prior, we wouldn't need a "minimal required exploration rate". The agent would just explore/exploit optimally according to the prior.
In theory, changing the exploration rate and changing the prior are equivalent. I think that it might be easier to decide upon an exploration rate that gives a good result for generic priors, than to be sure that generic priors have good exploration rates. But this is just an impression.
Many people (including me) had the impression that AIXI was ideally smart. Sure, it was uncomputable, and there might be "up to finite constant" issues (as with anything involving Kolmogorov complexity), but it was, informally at least, "the best intelligent agent out there". This was reinforced by Pareto-optimality results, namely that there was no computable policy that performed at least as well as AIXI in all environments, and strictly better in at least one.
However, Jan Leike and Marcus Hutter have proved that AIXI can be, in some sense, arbitrarily bad. The problem is that AIXI is not fully specified, because the universal prior is not fully specified. It depends on a choice of a initial computing language (or, equivalently, of an initial Turing machine).
For the universal prior, this will only affect it up to a constant (though this constant could be arbitrarily large). However, for the agent AIXI, it could force it into continually bad behaviour that never ends.
For illustration, imagine that there are two possible environments:
Now simply choose a language/Turing machine such that the ratio P(Hell)/P(Heaven) is higher than the ratio 1/ε. In that case, for any discount rate, the AIXI will always output "0", and thus will never learn whether its in Hell or not (because its too risky to do so). It will observe the environment giving reward ε after receiving "0", behaviour which is compatible with both Heaven and Hell. Thus keeping P(Hell)/P(Heaven) constant, and ensuring the AIXI never does anything else.
In fact, it's worse than this. If you use the prior to measure intelligence, then an AIXI that follows one prior can be arbitrarily stupid with respect to another.