"Transfinite towers" often run into the problem of termination: when $F$ is a "nice" function on ordinals (= monotonic, nondecreasing, and continuous), we get $$F(\sup\{F^n(0): n\in\omega\}=\sup\{F^n(0):n\in\omega\}.$$ That is, the "iterating $F$" tower stops at level $\omega$. Moreover, if the sequence $(F^n(0))_{n\in\omega}$ was strictly increasing, then this least fixed point of $F$ will have cofinality $\omega$ and so not be inaccessible.
The "$+1$"-versions of such towers avoid this termination issue, at the cost of being a bit stranger. For example, consider the function on ordinals defined recursively as follows:
This function never "stops," in the sense that we always have $S(\alpha)<S(\alpha+1)$. We can then try to "strip off" the added $+1$s by taking cardinalities: let $$\hat{T}(\alpha)=\vert S(\alpha)\vert.$$ This $\hat{T}$ function isn't quite your $T$ but it's fairly similar: it begins $$\hat{T}(0)=\aleph_0, \quad\hat{T}(1)=\vert \omega_{\omega+1}+1\vert=\aleph_{\omega+1},\quad\hat{T}(2)=\vert\omega_{\omega_{\omega+1}+1}+1\vert=\aleph_{\omega_{\omega+1}+1}, \quad...$$ In particular, we get $$T(0)=\hat{T}(0)<T(1)<\hat{T}(1)<T(2)<\hat{T}(2)<...$$ (Note that $\aleph$s should not be used in subscripts to $\aleph$ numbers.)
The functions $S$ and $\hat{T}$ are provably total in $\mathsf{ZFC}$, and - being "non-silly" (e.g. not having an inaccessible somehow baked in if possible) - don't reach up to an inaccessible before we feed in an inaccessible at the outset. Specifically, the least $\alpha$ such that $S(\alpha)$ is $\ge$ the least inaccessible (in fact, equal to the least inaccessible $+1$) is the least inaccessible itself.
But from the end of your question, it sounds like this isn't quite what you want; rather, you want an iterative process which does have a fixed point, but whose least fixed point is extremely large (e.g. plausibly an inaccessible cardinal). This is going to be tricky: since $\mathsf{ZFC}$ doesn't prove that inaccessibles exist (unless it's inconsistent in the first place!), such a function has to have some aspect which is independent of $\mathsf{ZFC}$.
Here's one example of such a function, albeit in a modified setting. Working in $\mathsf{ZFC}$ + "There is a proper class of inaccessible cardinals," and restricting to uncountable cardinals for simplicity, let $C(\kappa)$ be the smallest inaccessible cardinal $\lambda$ such that every unsatisfiable $\mathcal{L}_{\kappa,\kappa}$-theory of size at most $\kappa$ has an unsatisfiable subset of size $<\lambda$. Obviously $C(\kappa)\le\kappa^+$; fixed points for $C$ are exactly the weakly compact cardinals, whose existence is independent of $\mathsf{ZFC}$ + "There is a proper class of inaccessible cardinals."
But the function $C$ itself isn't really very interesting, and in fact the property of being a $C$-fixed point is more naturally expressed without reference to $C$ itself (namely, "is inaccessible and has the weak compactness property"). In general, large cardinals are rarely best thought of as least fixed points.
