Recurrence relations that do not have a closed form solution
Yes, there are. Consider the recurrence $a_{n+1}=(7a_n/4 + 1/2) - (5a_n/4 + 1/2)(-1)^{a_n}$ where $a_0$ is a given integer value. It is well known that this gives the Collatz sequence. There is no "known" closed form solution to the recurrence (and if you find it, you would be quite famous in mathematical circles). Furthermore, it is also well known that there are recurrent sequences that can't be solved in closed form. For instance, Conway shows that a natural generalization of the Collatz problem is undecidable. On the other hand, there are large classes of recurrence relations for which methods are known to determine closed form solutions in terms of sometimes multiple finite or infinite sums. Hypergeometric functions can be used to solve large classes of them. For instance, in theory, all linear recurrences with polynomial coefficients can be solved. More information can be found in Solving Linear Recurrence Equations with
Polynomial Coefficients by Petkovsek and Zakrajsek (PDF).