What is the least upper bound for the size of an increasing chain in $(\omega^\omega,<^*)$? Since every family of functions $F\subseteq\omega^\omega$ of size $\left|F\right|<\mathfrak{b}$ is bounded, one can easily define a sequence $\{f_\alpha:\alpha<\kappa\}$ such that $\kappa=\mathfrak{b}$ and $f_\alpha<^* f_\beta$ for $\alpha<\beta<\kappa$.
What is the least upper bound for the size $\kappa$ of such sequence? Can one have $\kappa=\mathfrak{c}$?
By $f<^*g$ I mean $(\exists k)(\forall n>k)\ f(n)<g(n)$.
 A: Claim. Assume GCH and let $\kappa > \omega_1$ be any regular uncountable cardinal. Then, after forcing with the $\kappa$-length finite support product of Cohen forcing, $\mathbb{C}_{\kappa}$, there is no chain of length greater than $\omega_1$.
This is proved via a so called "isomorphism of names" argument. Unfortunately I do not have a reference for this (maybe somebody else does), but I will outline the argument.
Proof. Assume $\dot f_\alpha$, $\alpha < \lambda$, are nice $\mathbb{C}_\kappa$-names for elements of $\omega^\omega$, that purportedly will form a chain and $\omega_1 < \lambda \leq \kappa$, $\lambda$ regular.
A nice name is, as usual, a name of the form $\bigcup_{s \in \omega^{<\omega}} \{ \check s \} \times A_{s}$, where each $A_s$ is an antichain. Thus let us write $\dot f_\alpha = \bigcup_{s \in \omega^{<\omega}} \{ \check s \} \times A_{s,\alpha}$ for each $\alpha < \lambda$. Then to each $\alpha < \lambda$, we can associate a "support" $S_\alpha \subseteq \kappa$, which is the union of all the supports of conditions in $A_{s,\alpha}$ for $s \in \omega^{<\omega}$. Then $S_\alpha$ is clearly countable and the evalutation of $\dot f_\alpha$ only depends on the restriction of the generic to $S_\alpha$. In fact, there is a Borel function $F_\alpha \colon (2^\omega)^{S_\alpha} \to \omega^\omega$ so that if $G = \langle x_i : i < \kappa \rangle$ is the generic sequence, then $\dot f_\alpha [G] = F_\alpha(\langle x_i : i \in S_\alpha \rangle)$.
Next we apply the Delta-system Lemma (using GCH), to get $X \in [\lambda]^\lambda$ and a root $R$ so that for every $\alpha \neq \beta \in X$, $S_\alpha \cap S_\beta = R$.
Now the point is, that by a pigeonhole principle (using CH and the fact that there are only continuum many Borel functions) there are $\alpha < \beta$ so that $F_\alpha$ and $F_\beta$ are the same modulo a permutation of $\kappa$ mapping $S_\alpha$ to $S_\beta$ fixing $R$ and everything else. This permutation induces an automorphism $\pi \colon \mathbb{C}_\kappa \to \mathbb{C}_\kappa$, that swaps $\dot f_\alpha$ and $\dot f_\beta$. But then, since it is forced that $\dot f_\alpha <^* \dot f_\beta$, we must also force that $\dot f_\beta <^* \dot f_\alpha$, which is impossible. This relies on the following well-known fact (I'm sure you can find it in Jech):
Fact. If $\varphi(x_0, \dots, x_n)$ is a formula in the language of set theory, $\tau_0, \dots, \tau_{n}$ names and $\pi$ and automrphism of the poset, then $$\Vdash \varphi(\tau_0, \dots, \tau_n) \text{ iff}\Vdash \varphi(\pi(\tau_0), \dots, \pi(\tau_n)).$$
Here we apply this fact to a formula $\varphi(x_0,x_1)$ expressing that $x_0, x_1 \in \omega^\omega$ and $x_0 <^* x_1$. $\square$
It is crucial that $<^*$ is definable without parameters (or definable over the ground model), so that its meaning is not changed by applying the automorphism. Compare this for example with a well-order of the reals in the extension. We can do everything up to the last step, where we can not apply the autormorphism argument since the well-order is not definable. (This is exactly the proof that there is no such definable well-order that you can find in textbooks!)
Of course this argument is very general and applies to basically any relation other than $<^*$ that is definable over the ground model.
Now how to get other values than $\omega_1$? Well first just force $\mathfrak{b}$ what ever you like to get a chain. That chain will of course be preserved in any ccc extension. Then run the argument above again replacing $\omega_1$ with the value of the continuum in that model.
This seems to only miss the case where the lub for the size of a chain could be singular. But who knows anything about singular cardinals?
