Let us consider the set $\omega^\omega$ of all maps $\omega\to\omega$ with the pointwise ordering. By cofinality of $(\omega^\omega,\le)$ I mean the smallest cardinality of the subfamily $\mathcal B$ such that for each $f\in\omega^\omega$ there is a $g\in\mathcal B$ such that $f\le g$, i.e., $\mathcal B$ is cofinal.
I understand that every cofinal family in $(\omega^\omega,\le)$ is dominating. (A dominating family is basically the same thing as cofinal in $(\omega^\omega,\le^*)$, perhaps with the slight distinction that we are talking about equivalence classes, when dealing with $\le^*$.) I do not think that the opposite implication is true (i.e., dominating $\Rightarrow$ cofinal).
Is it known whether cofinality of $(\omega^\omega,\le)$ is $\mathfrak d$? (Here $\mathfrak d$ denotes the dominating number.)
It is true that the cofinality of $(\omega^\omega,\leq)$ is $\mathfrak{d}$. While a dominating family $\mathcal{D}$ might not be cofinal in this sense, we can obtain a cofinal family from it by taking all finite modifications of the functions in $\mathcal{D}$, expanding all of the equivalence classes of these functions, as it were. The new family $$\mathcal{D}'=\{f;\exists g\in\mathcal{D}\colon g=f\text{ almost everywhere}\}$$ is a cofinal family in $(\omega^\omega,\leq)$ and clearly has the same cardinality as $\mathcal{D}$.