It is well known, that the Clifford $+T$ gate set consisting of the gates $\lbrace H, S, CNOT, T \rbrace$ is universal for quantum computation, that is, for any n-qubit unitary $U:\left( \mathbb{C}^2\right)^{\otimes n} \rightarrow \left( \mathbb{C}^2\right)^{\otimes n}$ and for every $\epsilon >0$, there is a unitary $V$ composed of gates from the Clifford $+T$ gate set such that $$\|U |0\rangle ^{\otimes n} - V|0\rangle ^{\otimes n}\|<\epsilon$$
Let now $\mathcal{C} = \lbrace V |0\rangle ^{\otimes n}: V \text{ composed from Clifford }+T \text{ gates}\rbrace$ and define the $T$-count $\tau(|c\rangle )$ of an element $|c\rangle \in \mathcal{C}$ as the minimal number $r$ such that there is a quantum circuit $V$ composed of arbitrarily many Clifford gates and $r$ $T$-gates such that $ |c\rangle = V |0\rangle ^{\otimes n}$.
The $T$-count can be arbitrarily large: Clearly, the universality of $\mathcal{C}$ implies that $\mathcal{C}$ is infinite. But since the Clifford group is (up to prefactors of the form $e^{i\theta}, \theta \in [0,2\pi]$) finite, we only reach (up to prefactors) a finite number of states with a finite number of $T$-gates.
Now my question: Do we know explicit families of states which have high $T$-count? An example for what I am looking for can be found on page 18 of this paper. There, the authors show - using the stabilizer nullity as a monotone - that for the controlled $Z$ gates with $n-1$ controls $C^nZ$, the state $C^nZ |+\rangle ^{\otimes n}$ has $T$-count $\Omega(n)$ (it is well known that $C^nZ |+\rangle ^{\otimes n}\in \mathcal{C}$, see e.g. Nielsen Chuang). Another example would probably be $|T\rangle^{\otimes n}$, where $|T\rangle = T |+\rangle $ (and all obvious variations of this).
I wonder if there are explicit examples where the $T$-count scales stronger with $n$. For example, I would be interested in seeing an explicit example of a family of states $|\Psi_n\rangle \in \left( \mathbb{C}^2\right)^{\otimes n}$ such that $\tau (|\Psi_n\rangle) = \Omega(f(n))$ where $f$ is maybe superlinear or even exponential.
(Maybe an interesting related observation: One will not be able to show something like this using the stabilizer nullity, since the stabilizer nullity is always bounded by the number of qubits!)
Alternatively: As the $T$-count can be arbitrarily large even for one qubit unitaries by the above argument, one can of course also ask the following (stronger) variant: Can we explicitly give a family $|\Psi_n\rangle \in \mathbb{C}^2$ such that $\tau(\Psi_n) > \tau(\Psi_{n-1})$ for all $n\in \mathbb{N}$?
I am quite new to this part of quantum information theory, any kind of insights are welcome - and also any kind of comment on my introductory thoughts.