Low for EXP and NEXP What are the largest classes which are low for EXP and NEXP?

For example: I am aware the class P, QP are low for EXP as well as NEXP. We also know that NP is not low for either of them. 

Is class BPP low for EXP or NEXP? What is the largest oracle class we can provide to EXP (or NEXP) which doesn't increase its power?
$\mathsf{BPP=NEXP}$ is consistent with our current knowledge, thus, since $\mathsf{EXP, NEXP}$ are not low for themselves, it is not known whether $\mathsf{BPP}$ is low for them. A positive answer would imply $\mathsf{BPP\neq NEXP}$, and a negative answer would imply $\mathsf{P\neq BPP}$ (and in turn, that $\mathsf{P\neq NP}$), so currently, if you're looking for unconditional results, you will have no luck with $\mathsf{BPP}$.

Since even a (possibly weak) class as $\mathsf{BPP}$ is not known to be low for $\mathsf{EXP}$, you will need to look between $\mathsf{P}$ and $\mathsf{BPP}$ (which does not give you much wiggle room), or for classes who are currently incomparable with $\mathsf{BPP}$ (such as $\mathsf{QP}$ as you yourself noted, I doubt you can find a significantly stronger low class for EXP).