This reminds me of a question I had seen on both MO and MSE. Sequence A276175 in the OEIS is defined by $$a_n = \frac{(a_{n-1} + 1)(a_{n-2}+1)(a_{n-3} + 1)}{a_{n-4}}$$ with $a_0 = a_1 = a_2 = a_3 = 1$. The OEIS page conjectures it to be an integer for all $n$. The MSE question contains a proof the all $a_n$ are integer (though I haven't read the proof). In the comments of the MO question it is observed $a_8$ is not Laurent.

This reminds me of a question I had seen on both MO and MSE. Sequence A276175 in the OEIS is defined by $$a_n = \frac{(a_{n-1} + 1)(a_{n-2}+1)(a_{n-3} + 1)}{a_{n-4}}$$ with $a_0 = a_1 = a_2 = a_3 = 1$. The OEIS page conjectures it to be an integer for all $n$. The MSE question contains a proof the all $a_n$ are integer (though I haven't read the proof). In the comments of the MO question it is observed $a_8$ is not Laurent.

Added in edit: I offer another example which does not exhibit the Laurent phenomenon, but conjecturally is an integer sequence. Consider the sequence defined by $$b_n = \frac{(b_{n-1} + 1)(b_{n-2} + 1)(b_{n-3} + 1)(b_{n-4}+1)}{b_{n-5}}$$ where $b_0 = b_1 = b_2 = b_3 = b_4 = 1$. I computed and after reducing I found the denominator of $b_{10}$ to be $b_0^14(b_1 + 1)b_1^8(b_2 + 1)b_2^4b_3^2b_4$ (not a monomial). I verified this sequence to be integer up to $n=36$, and Kevin O'Bryant later verified up to $n=41$. I asked if the sequence is integer for all $n \geq 0$ in a separate question.