Suppose your sequence has a limit, say L.

Then $\lim_{n\to \infty}a_n=\lim_{n\to \infty}a_{n+1}=L$ and the recurrence relationship implies that L=1+$\frac{1}{1+L}$ which in term implies that
(L+1)(L-1)=1 $\Rightarrow L^2-1=1 \Rightarrow L=\sqrt{2}$ or $L=-\sqrt{2}$ , assuming $1+L\neq-1$.

Now, notice that your sequence is bounded below by 1, bounded above by 2 and Cauchy, so it has a positive limit.

Indeed, let $n,m\in\mathbf{N} $ with $n\lt m$. Then $$|a_{m+1}-a_{n+1}|= |\frac{a_n-a_m}{(1+a_{m})(1+a_{n})}|=|\frac{a_{n-1}-a_{m-1}}{(1+a_{m})(1+a_{n})(1+a_{m-1})(1+a_{n-1})}|$$ and inductively you have that $|a_{m+1}-a_{n+1}|\leq|\frac{1}{2^n}|$ , with $lim_{m,n\to \infty}|\frac{1}{2^n}|=0$ , because $|a_{i-1}-a_{j-1}|\leq1$ and $(1+a_i)\geq2$ for every $i,j\in\mathbf{N}$.