In his article Foncteurs analytiques et espèces de structures (Lecture Notes in Mathematics 1234), Joyal characterizes analytic functors as those which preserve filtered colimits, cofiltered limits, and weak pullbacks. So it's just a matter of finding functors which violate one of these properties.

A functor that preserves filtered colimits is said to be finitary, and it is easy to come up with examples of non-finitary endofunctors on $\mathbf{Set}$. One example is the covariant power set functor $P$; this does not preserve for example the filtered diagram $D$ of finite subsets of $\mathbb{N}$ and inclusions between them, by simple cardinality considerations. Another is the endofunctor $F = (-)^\mathbb{N}$, where again the comparison map $\text{colim} \; F D \to F(\mathbb{N})$ is not an isomorphism (the identity map $1_\mathbb{N}$ as an element of $F(\mathbb{N})$ is not in the image of the comparison map).

In his article Foncteurs analytiques et espèces de structures (Lecture Notes in Mathematics 1234), Joyal characterizes analytic functors as those which preserve filtered colimits, cofiltered limits, and weak pullbacks. So it's just a matter of finding functors which violate one of these properties.

A functor that preserves filtered colimits is said to be finitary, and it is easy to come up with examples of non-finitary endofunctors on $\mathbf{Set}$. One example is the covariant power set functor $P$; this does not preserve for example the colimit of the filtered diagram $D$ of finite subsets of $\mathbb{N}$ and inclusions between them, by simple cardinality considerations. Another is the endofunctor $F = (-)^\mathbb{N}$, where again the comparison map $\text{colim} \; F D \to F(\mathbb{N})$ is not an isomorphism (the identity map $1_\mathbb{N}$ as an element of $F(\mathbb{N})$ is not in the image of the comparison map).