  We introduce infinitary action logic with exponentiation -- that is, the
multiplicative-additive Lambek calculus extended with Kleene star and with a
family of subexponential modalities, which allows some of the structural rules
(contraction, weakening, permutation). The logic is presented in the form of an
infinitary sequent calculus. We prove cut elimination and, in the case where at
least one subexponential allows non-local contraction, establish exact
complexity boundaries in two senses. First, we show that the derivability
problem for this logic is $\Pi_1^1$-complete. Second, we show that the closure
ordinal of its derivability operator is $\omega_1^{\mathrm{CK}}$. In the case
where no subexponential allows contraction, we show that complexity is the same
as for infinitary action logic itself. Namely, the derivability problem in this
case is $\Pi^0_1$-complete and the closure ordinal is not greater than
$\omega^\omega$.
