Define a growth function to be a monotone increasing function $F: {\bf N} \to {\bf N}$, thus for instance $n \mapsto n^2$, $n \mapsto 2^n$, $n \mapsto 2^{2^n}$ are examples of growth functions.  Let's say that one growth function $F$ dominates another $G$ if one has $F(n) \geq G(n)$ for all $n$.  (One could instead ask for eventual domination, in which one works with sufficiently large $n$ only, or asymptotic domination, in which one allows a multiplicative constant $C$, but it seems the answers to the questions below are basically the same in both cases, so I'll stick with the simpler formulation.)  

Let's call a collection ${\mathcal F}$ of growth functions complete cofinal if every growth function is dominated by at least one growth function in ${\mathcal F}$.  

Cantor's diagonalisation argument tells us that a cofinal set of growth functions cannot be countable.  On the other hand, the set of all growth functions has the cardinality of the continuum.  So, on the continuum hypothesis, a cofinal set of growth functions must necessarily have the cardinality of the continuum.

My first question is: what happens without the continuum hypothesis?  Is it possible to have a cofinal set of growth functions of intermediate cardinality?

My second question is more vague: is there some simpler way to view the poset of growth functions under domination (or asymptotic domination) that makes it easier to answer questions like this?  Ideally I would like to "control" this poset in some sense by some other, better understood object (e.g. the first uncountable ordinal, the nonstandard natural numbers, or the Stone-Cech compactification of the natural numbers).

EDIT: notation updated in view of responses.

For asymptotic domination, commonly denoted ${\leq^*}$ and often called eventual domination, this has been answered by Stephen Hechler, On the existence of certain cofinal subsets of ${}^{\omega }\omega$, MR360266. What you call a complete set is usually called a dominating family. 

As a poset under eventual domination, a dominating family $\mathcal{F}$ must have the following three properties:


$\mathcal{F}$ has no maximal element.
Every countable subset of $\mathcal{F}$ has an upper bound in $\mathcal{F}$.
$|\mathcal{F}| \leq 2^{\aleph_0}$


Hechler showed that for any abstract poset $(P,{\leq})$ with these three properties, there is a forcing extension where all cardinals and cardinal powers are preserved, and there is a dominating family isomorphic to $(P,{\leq})$. 

In particular, one can have a wellordered dominating family whose length is any cardinal $\delta$ with uncountable cofinality. In this case, the restriction $\delta \leq 2^{\aleph_0}$ is inessential since one can always add $\delta$ Cohen reals without affecting conditions (1) and (2). However, for arbitrary posets, condition (2) could be destroyed by adding reals.

The total domination order is more complex. One can always get a totally dominating family $\mathcal{F}'$ from a dominating family $\mathcal{F}$ by adding $\max(f,n) \in \mathcal{F}'$ for every $f \in \mathcal{F}$ and $n < \omega$. Since $\mathcal{F}$ is infinite, the resulting family $\mathcal{F}'$ has the same size as $\mathcal{F}$. Howerver, there does not appear to be a simple combinatorial characterization of the possibilities for the posets that arise in this way.
