Do all complexity classes have a leaf language characterization?
Have a look at


  Bernd Borchert, Riccardo Silvestri: A Characterization of the Leaf Language Classes. Inf. Process. Lett. 63(3): 153-158 (1997) (doi link here)


The authors characterize the leaf language classes as those which are (a) "countable", (b) are "downward" closed wrt polytime many-one reducibility, and (c) "join-closed" (i.e., disjoint union) wrt polytime many-one reducibility. 

More formally, all the languages $L$ in a leaf language class have a bijection with the natural numbers, and the property that for every $C,D \in L$, if $E \leq^P_m C \sqcup D$ then $E \in L$ as well (the $\sqcup$ denotes disjoint union). Also, every "non-leaf language class" contains a language which fails to have one of these properties.

From these three conditions we can get many examples of classes which aren't leaf language classes. For example, the "countable" condition rules out advice classes like $P/poly$, and the "downward closed wrt polytime many-one reducibility" rules out fixed resource-bound classes like $SPACE[n]$. (Recall that the usual proof that $SPACE[n] \neq P$ uses the fact that $SPACE[n]$ is not closed under such reductions.)