Have I discovered a new significance to a previously discovered constant? I've been interested in infinite sums for a while, though I have no formal education of them. I was messing around with repeated division and addition (e.g. 1 + (1 / (1 + (1 /...)))) I then plugged fibonacci numbers into the above pattern, and as I calculated more and more layers, the result converged to around 1.39418655, which I just now found out was a constant called Madachy's constant.  However, from the research that I did on the constant (I found very little), it doesn't seem to be related to infinite series or Fibonacci numbers at all. Have I found a new way to calculate this number? Does it give it any more significance?
Did you see this? fq.math.ca/Scanned/6-6/madachy.pdf It looks like it is exactly your calculation.

Friendly edit: This is in Fibonacci Quarterly, Volume 6 No. 6 page 385, as seen at the cover of that issue.
The continued fraction you are calculating will converge to the golden ratio. It is most certainly a significant constant having history spanning several millennia.