Assume that such functions exist. Since the functions are increasing, by simple induction we get $ f(n),g(n)\ge n$ for all $ n\in\mathbb{N}$. If $ f(n)=n$ for some $ n\in\mathbb{N}$, subsituting to the given inequality, we get $ g(g(n))<g(n)$, contradicting the hypothesis that $ g$ is increasing. Similarly, if $ g(n)=n$ for some $ n\in\mathbb{N}$, we get $ f(n)<f(n)$ which is impossible. Therefore $ f(n),g(n)>n$.
We now have $ f(g(g(n)))<g(f(n))<f(g(f(n)))$. We get $ f(g(g(n)))<f(g(f(n)))$, so $ g(n)<f(n)$. Now, $ f(g(g(n)))<g(f(n))<f(f(n))$, so $ g(g(n))<f(n)$. By induction, we get that $ f(n)>g(g(...(n)...))$. This is impossible because $ g(n)<g(g(n))<...<g(g(...(n)...))$. Therefore no such functions exist.