Let $\{\omega_i\}_{i\ge 1}$ be a bounded sequence of positive numbers and consider the following series of nested integrals $$ S=\sum_{n=1}^\infty \int_0^{t} \cos(\omega_1t_1)\left(\int_0^{t_1} \cos(\omega_2t_2)\cdots \left(\int_0^{t_{n-1}} \cos(\omega_nt_n)\, \mathrm{d}t_n\right)\cdots\mathrm{d}t_2\right) \mathrm{d}t_1 $$
If $\omega_i=\bar \omega$ for all $i$, then the above series converges to (see, e.g., this question) $$ \exp\left({\int_0^t \cos(\bar \omega \tau )\, \mathrm{d}\tau}\right)-1, $$ so that, in this special case, a simple upper bound to the series $S$ is given by $$ S\le \exp\left({\frac{1}{\bar \omega}}\right)-1. $$
Now, let's go back to the general case of different $\omega_i$'s and define $$\bar \omega:=\min_{i\ge 1} \omega_i.$$ Is it true that $$ S\le \exp\left({\frac{1}{\bar \omega}}\right)-1 \ \ \ ? $$