This is a question regarding two famous cardinal characteristics:
$\mathfrak{b}$ the minimal cardinality of an unbounded family in $(\omega^\omega,{<^*})$.
$\mathfrak{d}$ the minimal cardinality of a cofinal family in $(\omega^\omega,{<^*})$.
It is not difficult to show that a scale exists if and only if $\mathfrak{b} = \mathfrak{d}$, and then the minimal length of a scale is the common value of these two cardinal characteristics (which is also the cofinality of any scale). However, this equality need not hold. In fact, the only inequalities that must hold are $$\aleph_1 \leq \mathfrak{b} = cf(\mathfrak{b}) \leq cf(\mathfrak{d}) \leq \mathfrak{d} \leq \mathfrak{c}.$$ Using Hechler's Theorem (see this answer of mine) any pattern of cardinals consistent with the above is possible to attain by a forcing. See Andreas Blass's handbook of set theory chapter (available here) for more on this topic.