Because I can't find a function $h$ with the property $$h^{\circ 2}(x)=x^2+1$$ I'm looking for a function that almost has that property - that is, I would like to find a closed-form (and preferably elementary) function $h$ satisfying $$\lim_{x\to\infty} (x^2+1-h(h(x)))=0$$ or, equivalently, $$h(h(x))=x^2+1+\mathcal O(\epsilon(x))$$ where $\lim_{x\to\infty} \epsilon(x)=0$. But I haven't been able to do this either. I've tried functions in the form $$|x|^{\sqrt 2}+C$$ but none of them have worked. Can anybody find such a function $h$?

Because I can't find a function $h:\mathbb R\mapsto\mathbb R$ with the property $$h^{\circ 2}(x)=x^2+1$$ I'm looking for a function that almost has that property - that is, I would like to find a closed-form (and preferably elementary) function $h:\mathbb R\mapsto\mathbb R$ satisfying $$\lim_{x\to\infty} (x^2+1-h(h(x)))=0$$ or, equivalently, $$h(h(x))=x^2+1+\mathcal O(\epsilon(x))$$ where $\lim_{x\to\infty} \epsilon(x)=0$. But I haven't been able to do this either. I've tried functions in the form $$|x|^{\sqrt 2}+C$$ but none of them have worked. Can anybody find such a function $h$?