Is there a closed-form solution for the recurrence relation $x_{t+1} = x_{t}^\alpha + 1$, where $\alpha \in (0,1)$? If it were $x_{t+1} = x_{t}^\alpha$, then taking the Log of both sides would make it linear and easy to solve. However, the same thing does not work in the example. Taking the exponent of both sides does not help to linearize it either.
The question is relevant to a growth model in mathematical economics. One can think of $x_t$ as the amount of capital at time $t$ and $1$ as a resource. Having a closed-form solution for $x_t$ would greatly simplify working with growth models. More specifically, it would allow doing comparative statics in a more complicated model where the growth model is embedded. While some results can be proven without a closed-form solution, it would help in creating an important special case.