What is the height (or depth) of $[\mathbb{N}]^\infty$? (This question assumes familiarity with combinatorial cardinal characteristics of the continnum.)
Let $[\mathbb{N}]^\infty$ be the family of infinite subsets of $\mathbb{N}$,
partially ordered by $\subseteq^*$, where $a\subseteq^* b$ means $a\setminus b$ is finite.
Let $\mathfrak{ht}$ (sometimes called $\operatorname{Depth}^+([\mathbb{N}]^\infty)$) be the minimal cardinal number $\kappa$ such that there is no $\subset^*$-decreasing $\kappa$-sequence in $[\mathbb{N}]^\infty$.
Then $\mathfrak{t}<\mathfrak{ht}\le\mathfrak{c}^+$.
A classic result from Kunen's thesis asserts that, when adding $\kappa\ge\aleph_1$ Cohen reals to a model of CH, we obtain $\mathfrak{ht}=\aleph_2$.
Open-ended question. Can the hypothesis $\mathfrak{b}<\mathfrak{ht}$ be expressed using cardinal characteristics other than $\mathfrak{ht}$?
It is easy to see that 
"$\mathfrak{t}=\mathfrak{b}$ or $\mathfrak{b}<\mathfrak{d}$"
implies 
"$\mathfrak{b}<\mathfrak{ht}$". 
Question 1. Is it consistent that 
"$\aleph_1=\mathfrak{t}<\mathfrak{b}=\mathfrak{c}=\aleph_2<\mathfrak{ht}$"?
In the Laver model, we have 
$\aleph_1=\mathfrak{t}<\mathfrak{ht}=\mathfrak{b}=\mathfrak{c}=\aleph_2$.
Question 2. What is the value of $\mathfrak{ht}$ in the Hechler and Mathias models?
Update: Will Brian answers Question 1 in the positive below, in the Hechler model (thus also answering Question 2 for Hechler). 
I thought that $\mathfrak{b}<\mathfrak{ht}$ implies there is a nontrivial set of reals satisfying the selection principle $\operatorname{S}_1(\Gamma,\Gamma)$, by
the the linked paper. But Brian's comments make it clear that I oversimplified the question for this purpose. I opened a new question that fits better the intended application. 
 A: In the Hechler model, $\aleph_1 = \mathfrak{t} < \mathfrak{b} = \mathfrak{c} = \aleph_2 < \mathfrak{ht}$.
(By "the Hechler model" I mean the result of a length-$\omega_2$ finite support iteration of the forcing to adjoin a dominating real.)
That $\aleph_1 = \mathfrak{t} < \mathfrak{b} = \mathfrak{c} = \aleph_2$ is discussed in Section 11.6 of Blass's handbook article. That $\mathfrak{ht} > \mathfrak{c}$ follows from Theorem 4.1 in


J. Baumgartner and P. Dordal, "Adjoining dominating functions," Journal of Symbolic Logic vol. 50 (1985), pp. 94 - 101, available here.


Interestingly, Baumgartner and Dordal show that, while there are $\subset^*$-decreasing sequences of length $\omega_2$ in this model, none of them are maximal -- all maximal sequences have cofinality $\omega_1$.
This answers question 1 and the first part of question 2. I don't know what $\mathfrak{ht}$ is in the Mathias model, and I also don't know the answer to your open-ended question (though I think it's an interesting one).
