Quiz: Infinite Continued fractions

At the SouthWest Fox speaker dinner we were talking about continued (or infinite) fractions.

I posed this question:

What does this expression evaluate to?

       1 + 1

                  _________

                   1 + 1

                          ________

                           1 + 1

  _______

                                  1 + 1

                                         ______

                                          1 + 1

                                              …

Clue: It’s adding some positive value to 1, so it must be larger than one. The thing it’s adding is the reciprocal of some number larger than 1 so….