Are these continued fractions of integrals known?
Not an answer, but one more picture, with the functions for 100 and 150 terms superposed.
It is not clear what is happening on the left.
EDIT:
Here is the code (using sagemath):
def fun(x, N=150):
L = [x**k/gamma(k + 1) for k in range(N, -1, -1)]
resu = 1.0
for t in L:
resu = t / (1 + resu)
return resu
P1 = plot(lambda x:fun(x, 100),-12, 3, aspect_ratio=1, color='red'); P1