Automata learning without counterexamples
Consider password automata: for each $w\in\{0,1\}^n$, the DFA $M_w$ accepts the language $\{w\}$. In this case, a membership query is the same as an equivalence query --- and clearly, you'll need exponentially many of these to find the "needle in the haystack". (This is even if the learner knows in advance that the target automaton is of this form.) For a formal proof, a teacher could simply respond to the first $2^{n}-1$ membership/equivalence queries with NO, adaptively choosing the target automaton as the unique remaining possibility.

Update. I neglected to mention that the password automaton has $n+1$ states and is learnable using a single standard Angluin equivalence query (namely, with candidate $M$ as the empty automaton).