As I've noted in the comments, the Śleszyński-Pringsheim Theorem guarantees that this continued fraction converges to a finite value (since the prime numbers are always greater than or equal to $2$).

This arXiv preprint gives an approximate value $s$ for the continued fraction whose partial denominators are prime numbers:

$$s=0.43233208718590286890925379324199996370511089688\dots$$

which the author computed with PARI/GP. He notes that the Plouffe inverter does not at all recognize this constant.

Here's some *Mathematica* code for computing the constant to `prec`

or so significant figures:

```
prec = 500;
y = N[2, prec + 5]; c = y; d = 0; k = 2;
While[True,
p = Prime[k];
c = p + 1/c; d = 1/(p + d);
h = c*d; y *= h;
If[Abs[h - 1] <= 10^(-prec), Break[]];
k++];
N[1/y, prec]
```

which yields the result `0.43233208718590286890925379324199996370511089687765131032815206715855390511529588664247730234675307312901358874751711021925473474173059981681532525370102846860319246045704466728602248840679362020193843643798792955246786129609763893526940277522319731978458635595794036202066338633654544895108909659715862787332585763686200183679952128087865043794610126643260422526400822552675221511335417037835319471839444535578027072057047898703982387228841680143293913635134426277200532815721973910424896503203478507`