Simple Question about e^(1/e)

[bo198214]

What is the smallest value of X???

You mean: What is the infimum of x such that your sequence still converges?

Interesting question, I never heard about that, and honestly I can not imagine that there exist results about it.
One would have to start showing that it converges for x=3 and that it diverges for x=1.

Also there is slight difference, the loop that you give computes:
^(1.444 + 1/4^x) ^ (1.444 + 1/3^x) ^ (1.444 + 1/2^x)
because you start with j=0.
If you would start with j=1 what would produce the sequence you give:
^(1.444 + 1/4^x) ^ (1.444 + 1/3^x) ^ (1.444 + 1/2^x) ^ (1.444 + 1)
but probably a different value of x.