$a_{n+1}=a_n^{n+1}+a_n+1$

Let

$$x = \lim_{n\to\infty} a_n^{1/n!}.$$

We know from @frame95's answer that $x$ exists. Now set $\delta_n = \log a_n - n! \log x$ and notice that

$$ \frac{\delta_{n+1}}{(n+1)!} = \frac{\delta_n}{n!} + \frac{1}{(n+1)!}\log\left( 1 + \frac{1}{a_n^n} + \frac{1}{a_n^{n+1}} \right). $$

Since we know that $\delta_n/n! \to 0$, recursively applying the above relation and taking limit shows that

$$ \frac{\delta_n}{n!} = -\sum_{k = 0}^{\infty} \frac{1}{(n+k+1)!}\log\left( 1 + \frac{1}{a_{n+k}^{n+k}} + \frac{1}{a_{n+k}^{n+k+1}} \right) $$

Now by utilizing simple estimations $a_{n+k} \geq a_n^{k!} \geq a_1^{k!}$ and $\log(1+t) \leq t$ for $t \geq 0$, we obtain the following very crude bound

$$ \left| \delta_n \right| \leq \sum_{k=0}^{\infty} \frac{n!}{(n+k+1)!} \cdot 2 a_1^{-k!}. $$

Since each summand is bounded by $2a_1^{-k!}$ and $\sum_{k=0}^{\infty} 2a_1^{-k!} < \infty$, dominated convergence theorem tells that

$$ \limsup_{n\to\infty} \left| \delta_n \right| \leq \sum_{k=0}^{\infty} \lim_{n\to\infty} \frac{n!}{(n+k+1)!} \cdot 2 a_1^{-k!} = 0. $$

Therefore $\delta_n \to 0$ and hence $a_n!/x^{n!} = e^{\delta_n} \to 1$ as required.

The guess is wrong, but if you go a step further... Let's stop the chain of inequalities a bit before: $$ a_n> a_{n-1}^n > \ldots > a_2 ^{n!/2} = (\sqrt{a_2} )^{n!}$$ Is this a better estimate? Yes, because $ \sqrt{a_2 }=\sqrt{ a_1^2 + a_1+1} > a_1$. Let's call $d_r = a_r^{1/r!}$. Generalizing this argument, we get $$ a_{r+k} > a_{r+k-1}^{r+k} > \ldots > a_r^{ (r+k)!/r!} = d_r^{(r+k)!}$$ Let's analyze $\{d_r\}$. We will show that it is an increasing and bounded sequence, thus convergent to a number $\ell$. Infact $$ (1) \ \ increasing \ \ \ d_{r+1} = a_{r+1}^{1/(r+1)!} = ( a_r^{r+1} +a_r+1)^{1/(r+1)!} > d_r $$ By induction, one can easily show that $a_n +1 < (a_1+1) a_n^{n+1}$. Thus, $a_{n+1} < (a_1+2)a_n^{n+1}$. This yields $$ (2) \ \ bounded \ \ \ d_{n+1} < (a_1+2)^{1/(n+1)!} d_n < 2^{\sum^{n+1} (1/k!)} a_1/2 < 2^{e-1} a_1 $$ If $x$ exists, then $x=\ell$. Infact $$ \lim_{n \to \infty} \frac{a_n}{x^{n!}} = \lim_{n \to \infty} \left ( \frac{d_n}{x} \right )^{n!} = 1 \ \Rightarrow \ \frac{\ell}{x} = \lim_{n \to \infty} \frac{d_n}{x} = 1 $$ Here my results stop: I can't show the refined result that this $x$ exist!