calculus - $a_{n+1}=a_n+frac{1}{a_n}$ where $a_{1}=1$. Is $f(n)=a_n$ an elementary function?

Suppose $$a_{n+1}=a_n+\frac{1}{a_n}$$ where $a_{1}=1$. Is $f(n)=a_n$ an elementary function?

I haven't found any paper concerning it. Thanks for your attention!