How can I find an explicit expression for this recursively defined sequence?

The candidate limit value $\ell$, satisfying $$ \ell=1+\frac{1}{\ell}, $$ with solutions $\ell=\frac{1}{2}(1\pm\sqrt{5}) $, suggests a relation with the Fibonacci sequence $\{F_n\}$. In fact, if you put $$ u_n=\frac{F_{n+1}}{F_n}, $$ you have the recurrence relation $$ u_{n+1}=\frac{F_{n+2}}{F_{n+1}}=\frac{F_{n+1}+F_n}{F_{n+1}}=1+\frac{1}{u_n}. $$ Therefore, you can use all the information about the Fibonacci sequence. It is not necessary to repeat that here.

the initial term $u_0$ can not be $-1$ or $0$, so if the limit of $ u_n $ exist, then it is necessarily the limit $l$ is a positif root of the $l^2-l-1$, following it is $\frac{1+\sqrt{5}}{2}$.