The continued fraction for 
is [2; 1, 2, 5, 1, 1, 2, 1, 1, ...] (OEIS A002211).
A plot of the first 256 terms of the continued fraction represented as a sequence
of binary bits is shown above.
The convergents are 2, 3, 8/3, 43/16, 51/19, ... (OEIS A127005 and A127006).
The incrementally largest terms are 2, 5, 10, 24, 90, 770, ... (OEIS A054866), which occur at positions 0, 3, 10, 15, 23, 104, 1701, ... (OEIS A224852; illustrated above).
The plot above shows the positions of the first occurrences of 1, 2, 3, ... in the continued fraction, the first few of which are 1, 0, 9, 46, 3, 33, 75, 64, 118, 10,
103, 26, 102, 109, ... (OEIS A224851). The
smallest number not occurring in the first 
terms of the continued fraction are 236, 260, 265, 279,
282, ... (E. Weisstein, Jul. 22, 2013).
