Given an initial integer $x_0>0$, one can consider the first prime of the recursive sequence $x_i=1+2x_{i-1}$.

Naïvely such a prime should exist for $x_0$ arbitrary since the sequence $\log(x_i)$ is asymptotically an arithmetic progression. Sometimes it takes however some time: for $x_0=147$ my Maple algorithm stops when hitting a 771-digit number labelled as prime by Maple (Maple does not use a primality proof but some strong primality tests if I am not mistaken).

Starting with $x_0=658$ I lost patience: No prime among the first $56000$ iterates leading to numbers with almost $17000$ digits.

I tried to find an easy reason: If $x\longmapsto 2x+1$ is $k$-periodic for $k$ prime numbers $p_0,\dotsc,p_{k-1}$ such that $x_i\equiv 0\pmod{p_i}$ and $x_i>p_i$ then there is obviously no prime in this sequence. Such an easy argument fails for all small values of $k$ for the sequence $x_0=658,\dotsc$.

Perhaps my patience ran out a bit early and the sequence will hit eventually a prime:

Does an integral sequence given by $x_0>0$, $x_i=1+2x_{i-1}$ necessarily contain a prime number?