EXP-Complete Problems vs Subexponential Algorithms
Due to popular demand, I’m converting my comment to an answer.

A simple padding argument shows that for every constant $\epsilon&gt;0$, there exist EXP-complete problems in $\mathrm{DTIME}(2^{n^\epsilon})$. Indeed, fix an arbitrary EXP-complete problem $L$, and assume that it is computable in time $2^{n^c}$. Let $d&gt;c/\epsilon$, and consider the problem
$$L'=\left\{0^m\#w:w\in L,m\ge|w|^d\right\}.$$
On the one hand, $L$ is polynomial-time${}^\dagger$ reducible to $L'$ via the function $w\mapsto0^{|w|^d}\#w$, thus $L'$ is EXP-hard.

On the other hand, $L'$ is computable in time $2^{n^\epsilon}$: given an input of size $n$, we first check (in polynomial time) that it is of the form $0^m\#w$ for $m\ge n'^d$, where $n'=|w|$. Then we check if $w\in L$, which takes time $2^{n'^c}\le2^{m^{c/d}}\le2^{m^\epsilon}\le2^{n^\epsilon}$.



${}^\dagger$ Actually, the reduction as given is even uniform $\mathrm{AC}^0$, and it can be made DLogTime if we replace $|w|$ with an upper bound that is a power of two.