Real analysis – squeezing between polynomial and exponential

To let $ mathbb {R} _ + $ denote the set of positive reals. To let $ mathbb {R} _ + ^ { mathbb {R} _ +} $ denote the set of all functions $ f: mathbb {R} _ + an mathbb {R} _ + $, To the $ f, g: mathbb {R} _ + an mathbb {R} _ + $ we say $ f leq ^ * g $ If there is $ N in mathbb {N} $ so that $ f (x) leq g (x) $ for all $ x geq N $,

To let $ A = {f in mathbb {R} _ + ^ { mathbb {R} _ +}: n in mathbb {N}: f (x) = x ^ n text {for all x} in mathbb {R} _ + } $ and let it go $ B = {f in mathbb {R} _ + ^ { mathbb {R} _ +}: exists varepsilon in mathbb {R} _ +: f (x) = (1+ varepsilon )) ^ x text {for all x} in mathbb {R} _ + } $,

Question. Is there $ s in mathbb {R} _ + ^ { mathbb {R} _ +} $ so for all $ a in A $ and for everyone $ b in B $ we have $ a leq ^ * s leq ^ * b $?

Soft question. (Only out of interest, not required to accept the answer.) If the above main question is answered "yes," there is a "natural" problem that is not polynomial but solvable in "time." $ s $"