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We show that there is a set that is almost complete but not complete under polynomial-time many-one (p-m) reductions for the class E of sets computable in deterministic time 2lin. Here a set in a complexity class C is almost complete for C under some given reducibility if the class of the problems in C that do not reduce to this set has measure 0 in C in the sense of Lutz''s resource-bounded measure theory. We also show that the almost complete sets for E under polynomial time-bounded length-increasing one–one reductions and truth-table reductions of norm 1 coincide with the almost p-m-complete sets for E. Moreover, we obtain similar results for the class EXP of sets computable in deterministic time 2poly. [Copyright &y& Elsevier]