The union of NE, NPNE, NPNP^NE, and so on.
Is called "strong" to contrast it with the ordinary Exponential Hierarchy EH.
Note that we would get the same class if we replaced NE by NEXP.
There exists an oracle relative to which SEH is not contained in EH [Hem89].EH and SEH are incomparable for all anyone knows.
Has roughly the same relationship to E as PH does to P.
More formally, EH is defined as the union of E, NE, NENP, NE with Σ2P oracle, and so on.
See [Har87] for more information.
If coNP is contained in AM[polylog], then EH collapses to S2-EXP•PNP [SS04] and indeed AMEXP [PV04].
On the other hand, coNE is contained in NE/poly, so perhaps it wouldn't be so surprising if NE collapses.
There exists an oracle relative to which EH does not contain SEH [Hem89]. EH and SEH are incomparable for all anyone knows.