An easy diagonalization shows that for every countable family of functions $g_n:\mathbb{N}\to\mathbb{N}$, there is a function $f$ eventually exceeding any one of them. Just let $f(n)=\sup_{k\leq n}g_k(n)+1$.

Although it may seem difficult to extend this idea to uncountable families of functions, the fact is that it is consistent with the ZFC axioms of set theory that one may do so. Specifically, the *bounding* number $\frak{b}$ is the size of the smallest family of functions $f:\mathbb{N}\to\mathbb{N}$ which are not (eventually) bounded by any single function. The observation above shows that the bounding number is uncountable, and it is clearly at most continuum, so under the continuum hypothesis the bounding number is precisely $\aleph_1$. The interesting thing, however, is that the bounding number can be strictly larger than $\aleph_1$, and indeed, by the method of forcing, it can be made as large as you like.

There are numerous interesting set-theoretic issues arising in connection with the bounding number and the other cardinal characteristics of the continuum, some of which I explain in my answer to the MO question Is there a topology on growth rates of functions?. Perhaps the most related to this question are the bounding and dominating numbers, which provide two distinct concepts of measuring the height of the order.

These concepts are also mentioned in several other MO answers, such as here.