Literally, the class of ALL languages.
ALL is a gargantuan beast that's been wreaking havoc in the Zoo of late.
First [Aar04b] observed that PP/rpoly (PP with polynomial-size randomized advice) equals ALL, as does PostBQP/qpoly (PostBQP with polynomial-size quantum advice).
Then [Raz05] showed that QIP/qpoly, and even IP(2)/rpoly, equal ALL.
Nor is it hard to show that MAEXP/rpoly = ALL.
Also, per [Aar18], PDQP/qpoly = ALL.
On the other hand, even though PSPACE contains PP, and EXPSPACE contains MAEXP, it's easy to see that PSPACE/rpoly = PSPACE/poly and EXPSPACE/rpoly = EXPSPACE/poly are not ALL.
So does ALL have no respect for complexity class inclusions at ALL? (Sorry.)
It is not as contradictory as it first seems. The deterministic base class in all of these examples is modified by computational non-determinism after it is modified by advice. For example, MAEXP/rpoly means M(AEXP/rpoly), while (MAEXP)/rpoly equals MAEXP/poly by a standard argument. In other words, it's only the verifier, not the prover or post-selector, who receives the randomized or quantum advice. The prover knows a description of the advice state, but not its measured values. Modification by /rpoly does preserve class inclusions when it is applied after other changes.
The class of problems solvable by a syntactic BPP machine with O(log n) random advice bits whose probability distribution depends only on the input length n. For each n, there exists good advice such that the output is correct with probability at least 2/3.
Contains BPP/mlog. The inclusion is strict, because BPP/rlog contains any finitely sparse language by fingerprinting; see the discussion for ALL.
Contained in BPP//log.
The class that does not contain any languages. (It might not surprise you that I put this one in at the suggestion of a mathematician...)
Is the opposite of ALL, but does not equal the complement coALL = ALL.
Is closed under polynomial-time Turing reductions :-).
Equals SPARSE ∩ coSPARSE and TALLY ∩ coTALLY.