Example of recurrence relation without closed form expression?

Question

Can give an example of a recurrence relation for which there does not exist a closed form expression?

Answer 1

Let $p_1 = 2$ and let $p_i = \min\{x\in\mathbb{Z}\setminus\{1\} \;\colon p_j \nmid x \;\forall j\in [1,i-1] \}$ for $i \geq 2$.

Answer 2

The quadratic recurrence relation $x_{n+1} = r x_n (1 - x_n)$ (iterating the logistic map) exhibits chaotic behavior for various values of $r$, which at least rules out any straightforward closed forms.