"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:

• $S(0)=\omega+1$.

• $S(\alpha+1)=\omega_{S(\alpha)}+1$.

• $S(\lambda)=\sup\{S(\alpha): \alpha<\lambda\}+1.$

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; 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.

"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:

• $S(0)=\omega+1$.

• $S(\alpha+1)=\omega_{S(\alpha)}+1$.

• $S(\lambda)=\sup\{S(\alpha): \alpha<\lambda\}+1.$

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.