I am studying two sequence with their e.g.f. The first one are the Bell numbers (sequence $A000110$ on OEIS) defined as follows. $B_0=1$ and $$ B_n =\sum_{k=0}^{n-1} {n-1\choose k} B_k $$ They have a known e.g.f. (via OEIS) which is $$F_B(x)=e^{e^x-1}$$
The second one is the sequence A005046 on OEIS defined as follows $a_0=1$ and $$ a_n =\sum_{k=0}^{n-1} {2n-1\choose 2k} a_k $$
Clearly $a_n\ge B_n$ for every $n$.
However on OEIS it says that the e.g.f. of $(a_n)_{n\ge0}$ is $$ F_a(x)=e^{\cosh(x) - 1} =e^{\frac{e^x+e^{-x}}{2}-1} $$ which is less than $F_B(x)$ for $x>0$.
You are not comparing like with like:
The initial terms of the series for $e^{\exp(x) - 1}$ is $1 + \frac{x}{1!} + 2\frac{x^2}{2!} + 5\frac{x^3}{3!}+ 15\frac{x^4}{4!} +\cdots$ giving a sequence starting $1,1,2,5,15,\ldots$
But $e^{\cosh(x) - 1}$ is an even function, and in fact the e.g.f. for the number of partitions of an $n$-set into even blocks (rather than of a $2n$ block), so the series coefficients are only positive for even powers. The initial terms are $e^{\cosh(x) - 1} = 1 + \frac{x^2}{2!} + 4\frac{x^4}{4!} + 31\frac{x^6}{6!} +\cdots$ giving a sequence starting $1,1,4,31,\ldots$ for $2n$-blocks or starting $1,0,1,0,4,0, 31,\ldots$ for $n$-blocks.