A comment in A007018 a(n) = a(n-1)^2 + a(n-1), a(0)=1 claims

Subsequence of squarefree numbers (A005117). - Reinhard Zumkeller, Nov 15 2004

Is that really so?

As far as I know, it is an open problem if a polynomial $f \in \mathbb{Z[x]}$ of degree $\ge 5$ can be squarefree infinitely often (some sources require $f$ to be irreducible).

If the OEIS comment is correct, the sequence will give an infinite family of (irreducible) polynomials which are squarefree infinitely often.

Denote by $a_n$ the terms of OEIS A007018. Set $a_n = x$ and $$f(x)=a_{n+4}=x \cdot (x + 1) \cdot (x^{2} + x + 1) \cdot (x^{4} + 2 x^{3} + 2 x^{2} + x + 1) \\\\ \cdot (x^{8} + 4 x^{7} + 8 x^{6} + 10 x^{5} + 9 x^{4} + 6 x^{3} + 3 x^{2} + x + 1)$$

$f(a_n)=a_{n+4}$ will be squarefree infinitely often (including the irreducible degree 8 factor) and iterating $x \mapsto x^2+x$ will produce an infinite family of polynomials with this property.

Added For references of squarefree values of polynomials, the search terms are square free values of polynomials. E.g. here p.1 and here "11. Squarefree values of polynomials".