Asymptotics for a certain group of exponential generating functions

The exponential generating function for the sequence A143405 in the OEIS is exp(exp(x)(exp(x) - 1)). This paper analyzes the more general generating function exp(mexp(bx) + rexp(d*x) + s) and provides asymptotics for the sequences A143405, A355291, A002872, A002874 and others in the OEIS.