Let $X$ be the set of $\mathbb{R}$-valued sequences, i.e. $X := \mathbb{R}^{\mathbb{N}}=\{f: \mathbb{N} \to \mathbb{R}\}$, and let $S$ the set of sequences which can be expressed in closed form, i.e.: $$ S:= \{f \in \mathbb{R}^\mathbb{N} \space | \space f \text{ is in closed form}\} \subseteq X $$ Now since "closed form" is not well-defined: I basically mean the usual stuff. That is: $f$ is in closed form if there is a mathematical expression that can be evaluated in a finite number of operations. The allowed symbols in the expression are: constants, variables and applications of $!$ (factorial), $\exp, \ln$, the trigonometric and hyperbolic functions with their inverses, $\lfloor \cdot \rfloor, \lceil \cdot \rceil, [\cdot]$.
For example, $(a_k)_{k\in\mathbb{N}}\in S$ if $\displaystyle a_k = e^{(k!)^2\cdot \sin\left(\binom{2k}{k}\right)}$.
Now we have $S \subset X$, i.e. there are sequences which cannot be represented in closed form.
However, does at least the following statement hold? $$ \forall f \in X: \exists g \in S: f \leq g $$ (i.e. every sequence is bounded by a sequence which can be expressed in closed form)
I think your definition of closed form is equivalent to the primitive recursive functions. The Ackermann function eventually overtakes every primitive recursive function, so the answer is that no function in $S$ dominates it.
In particular, probably the fastest growing type of function in your library is $n^{n^{n^n}}$ for some finite height of the tower. Ackermann uses $2$'s instead of $n$'s, but makes the height increase without bound, so eventually it will be taller than whatever tower you pick.