I will first ask the question that can be asked very simply. Then I will explain a motivation and give references to related sequences.
Consider the sequence defined by
$$ b_n = frac {(b_ {n1} +1) (b_ {n2} + 1) (b_ {n3} + 1) (b_ {n4} +1)} {b_ {n5}} $$
or $ b_0 = b_1 = b_2 = b_3 = b_4 = $ 1.Question: is $ b_n in mathbb {Z} $ for everyone $ n geq 0 $?
Provided that my program is correct, I found that $ b_n in mathbb {Z} $ for $ 0 leq n leq $ 36.
I arrived at this reflection on MO323867 which requires sequences which have all values but do not satisfy the Laurent phenomenon.
More generally, we can consider the sequence defined by
$$ a ^ {(k)} _ n = frac { prod_ {i = 1} ^ {k1} (a ^ {(k)} _ {ni} + 1)} {a ^ {(k )} _ {nk}} $$
with $ a_0 = a_1 = cdots = a_ {k1} = $ 1.

When $ k = $ 2 we obtain $ a ^ {(2)} _ n a ^ {(2)} _ {n2} = a ^ {(2)} _ {n1} + 1 $ which can be seen coming from a guy $ A_2 $ Cluster algebra that implies the Lawrence phenomenon and completeness (see A076839).

When $ k = $ 3 we have the Laurent phenomenon and consequently the sequence is entirely (see A276123). The Laurent phenomenon follows for a more general result in Theorem 6.2.3 of Matthew Russell's thesis.

When $ k = $ 4 the sequence does not present the Laurent phenomenon, but it is always an entire sequence (see A276175, MO248604, MSE1905063).

When $ k = $ 5 we have the sequence of the question posed above.

When $ k> $ 5 the sequence does not consist solely of integers, the first violation being $ a ^ {(k)} _ {2k} not in mathbb {Z} $.
This can be seen by looking at the 2adic valuation. In particular, we can show $ nu_2 (a ^ {(k)} _ {2k}) = 5k $ for $ k geq $ 5. To see this, we can work $ pmod {2 ^ 5} $ and finds that $ (a ^ {(k)} _ {2k1} + 1) equiv 2 ^ 4 pmod {2 ^ 5} $. It's clear that $ a_k = 2 ^ {k1} $ and $ a_ {n} $ is even, equivalently $ (a_ {n} +1) $ is strange, for $ k leq n <2k – $ 1.
So the sequence $ b_n $ correspond to $ k = $ 5 is the only sequence of the family for which I do not know if all the terms are integers.