The simple continued fraction for $\pi$ begins

$$\pi = 3 + \frac{1}{{7} + \frac{1}{{15} + \frac{1}{{1} + \, \cdots}}}$$

and continues with the numerators 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, ... listed in A001203 of Sloane's OEIS.

Here are a few more terms: 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, 1, 1, 1, 3, 1, 1, 8, 1, 1, 2, 1, 6, 1, 1, 5, 2, 2, 3, 1, 2, 4, 4, 16, 1, 161, 45, 1, 22, 1, 2, 2, 1, 4, 1, 2, 24, 1, 2, 1, 3, 1, 2, 1, 1, 10, 2, 5, 4, 1, 2, 2, 8, 1, 5, 2, 2, 26, 1, 4, 1, 1, 8, 2, 42, 2, 1, 7, 3, 3, 1, 1, 7, 2, 4, 9, 7, 2, 3, 1, 57, 1, 18, 1, 9, 19, 1, 2, 18, 1, 3, 7, 30, 1, 1, 1, 3, 3, 3, 1, 2, 8, 1, 1, 2, 1, 15, 1, 2, 13, 1, 2, 1, 4, 1, 12, 1, 1, 3, 3, 28, 1, 10, 3, 2, 20, 1, 1, 1, 1, 4, 1, 1, 1, 5, 3, 2, 1, 6, 1, 4, 1, 120, 2, 1, 1, 3, 1, 23, 1, 15, 1, 3, 7, 1, 16, 1, 2, 1, 21, 2, 1, 1, 2, 9, 1, 6, 4, 127, 14, 5, 1, 3, 13, 7, 9, 1, 1, 1, 1, 1, 5, 4, 1, 1, 3, 1, 1, 29, 3, 1, 1, 2, 2, 1, 3, 1, 1, 1, 3, 1, 1, 10, 3, 1, 3, 1, 2, 1, 12, 1, 4, 1, 1, 1, 1, 7, 1, 1, 2, 1, 11, 3, 1, 7, 1, 4, 1, 48, 16, 1, 4, 5, 2, 1, 1, 4, 3, 1, 2, 3, 1, 2, 2, 1, 2, 5, 20, 1, 1, 5, 4, 1, 436, 8, 1, 2, 2, 1, 1, 1, 1, 1, 5, 1, 2, 1, 3, 6, 11, 4, 3, 1, 1, 1, 2, 5, 4, 6, 9, 1, 5, 1, 5, 15, 1, 11, 24, 4, 4, 5, 2, 1, 4, 1, 6, 1, 1, 1, 4, 3, 2, 2, 1, 1, 2, 1, 58, 5, 1, 2, 1, 2, 1, 1, 2, 2, 7, 1, 15, 1, 4, 8, 1, 1, 4, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 9, 1, 4, 3, 15, 1, 2, 1, 13, 1, 1, 1, 3, 24, 1, 2, 4, 10, 5, 12, 3, 3, 21, 1, 2, 1, 34, 1, 1, 1, 4, 15, 1, 4, 44, 1, 4, 20776, 1, 1, 1, 1, 1, 1, 1, 23, 1, 7, 2, 1, 94, 55, 1, 1, 2, 1, 1, 3, 1, 1, 32, 5, 1, 14, 1, 1, 1, 1, 1, 3, 50, 2, 16, 5, 1, 2, 1, 4, 6, 3, 1, 3, 3, 1, 2, 2, 2, 5, 2, 2, 2, 28, 1, 1, 13, 1, 5, 43, 1, 4, 3, 5, 3, 1, 4, 1, 1, 2, 2, 1, 1, 19, 2, 7, 1, 72, 3, 1, 2, 3, 7, 11, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 33, 7, 19, 1, 19, 3, 1, 4, 1, 1, 1, 1, 2, 3, 1, 3, 2, 2, 2, 2, 4, 1, 1, 1, 4, 2, 3, 1, 1, 1, 1, 11, 1, 1, 2, 1, 2, 1, 2, 2, 1, 7, 2, 27, 1, 1, 6, 2, 1, 9, 6, 26, 1, 1, 3, 2, 1, 1, 1, 1, 1, 15, 1, 36, 4, 2, 2, 1, 22, 2, 1, 106, 2, 2, 1, 3, 1, 12, 10, 7, 1, 2, 1, 1, 1, 1, 8, 2, 4, 5, 3, 2, 1, 4, 23, 1, 18, 2, 10, 3, 1, 6, 6, 13, 8, 6, 2, 2, 2, 2, 1, 1, 1, 3, 1, 7, 17, 1, 1, 1, 2, 5, 5, 1, 1, 2, 11, 1, 6, 1, 6, 1, 29, 4, 29, 3, 5, 3, 1, 141, 1, 2, 7, 7, 2, 2, 7, 1, 1, 7, 1, 7, 1, 2, 4, 1, 1, 1, 30, 1, 12, 4, 18, 10, 2, 8, 1, 2, 2, 2, 4, 13, 1, 5, 4, 1, 6, 1, 1, 11, 2, 4, 2, 1, 1, 3, 3, 12, 1, 1, 39, 5, 1, 1, 16, 125, 1, 4, 1, 2, 1, 19, 1, 4, 1, 1, 2, 1, 4, 1, 10, 1, 4, 2, 1, 1, 1, 5, 10, 4, 14, 1, 13, 41, 1, 4, 1, 8, 1, 1, 2, 1, 3, 1, 6, 1, 3, 2, 2, 2, 1, 4, 1, 14, 1, 2, 8, 1, 8, 3, 3, 3, 1, 37, 4, 2, 4, 1, 3, 4, 25, 4, 27, 2, 7, 1, 1, 2, 6, 1, 1, 1, 12, 1, 2, 2, 2, 13, 12, 1, 3, 1, 6, 1, 1, 33, 1, 5, 3, 1, 5, 15, 8, 8, 47, 1, 3, 2, 12, 2, 12, 1, 12, 1, 2, 5, 3, 1, 1, 1, 1, 2, 3, 5, 4, 2, 1, 1, 5, 1, 9, 14, 1, 1, 3, 2, 1, 9, 3, 22, 13, 1, 1, 3, 20, 1, 1, 61, 1, 376, 2, 107, 1, 10, 3, 2, 2, 31, 1, 2, 10, 2, 2, 62, 2, 2, 7, 4, 5, 6, 1, 1, 1, 1, 2, 8, 2, 73, 3, 5, 42, 1, 3, 2, 1, 1, 59, 6, 1, 1, 1, 5, 1, 6, 1, 2, 6, 1, 1, 1, 1, 3, 2, 1, 3, 1, 8, 1, 4, 2, 5, 4, 7, 1, 4, 2, 2, 6, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 5, 1, 2, 1, 1, 10, 1, 6, 1, 129, 1, 4, 65, 2, 4, 4, 3, 2, 3, 1, 1, 5, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 3, 1, 2, 1, 2, 4, 2, 1, 2, 27, 6, 2, 1, 193, 1, 3, 9, 1, 3, 35, 2, 1, 8, 1, 1, 1, 1, 9, 3, 56, 1, 6, 6, 2, 8, 1, 8, 1, 2, 3, 6, 3, 1, 3, 1, 1, 1, 2, 13, 1, 1, 1, 1, 13, 2, 1, 3, 1, 3, 15, 2, 1, 1, 2, 4, 1, 4, 5, 2, 2, 1, 2, 1, 6, 1, 4, 12, 1, 1, 1, 1, 13, 1, 3, 4, 1, 1, 1, 2, 9, 1, 7, 1, 1, 1, 1, 4, 1, 3, 4, 1, 1, 4, 3, 1, 39, 2, 1, 1, 1, 1, 1, 4, 7, 2, 2, 2, 1, 1, 1, 1, 2, 114, 12, 4, 1, 3, 2, 1, 19, 1, 1, 2, 1, 1, 3, 4, 1, 60, 3, 72, 2, 1, 1, 1, 50, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 4, 1, 7, 3, 1, 2, 1, 5, 1, 1, 1, 2, 6, 2, 21, 2, 6, 1, 6, 1, 1, 2, 1, 7, 1, 8, 1, 1, 5, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 11, 2, 4, 10, 2, 1, 1, 13, 1, 1, 7, 15, 1, 1, 1, 2, 3, 15, 8, 8, 2, 1, 13, 3, 5, 1, 2, 1, 6, 1, 10, 123, 3, 1, 4, 59, 4, 156, 88, 1, 5, 4, 1, 3, 1, 4, 2, 9, 1, 7, 4, 2, 1, 2, 3, 2, 1, 2, 11, 1, 13, 7, 7, 1, 63, 37, 12, 86, 1, 1, 1, 1, 2, 2, 4, 2, 18, 1, 1, 1, 41, 2, 1, 1, 12, 1, 2, 1, 1, 2, 10, 1, 1, 1, 5, 1, 1, 3, 1, 7, 5, 1, 9, 1, 2, 2, 7, 1, 1, 5, 2, 1, 3, 3, 5, 2, 1, 11, 3, 1, 3, 2, 1, 1, 2, 1, 14, 5, 2, 2, 1, 1, 1, 1, 3, 1, 3, 3, 2, 2, 1, 3, 2, 1, 2, 1, 4, 1, 14, 1, 1, 58, 7, 1, 2, 1, 1, 5, 1, 2, 1, 5, 18, 1, 4, 3, 1, 1, 1, 4, 1, 1, 2, 5, 1, 148, 1, 9, 2, 1, 2, 1, 5, 4, 93, 1, 1, 2, 4, 1, 2, 73, 1, 1, 3, 1, 1, 1, 1, 2, 1, 34, 1, 5, 6, 1, 2, 1, 3, 4, 1, 16, 28, 17, 2, 5, 5, 26, ...

Most computer algebra systems provide relatively simple commands for obtaining as many terms as one wants. In Mathematica, for example, one can issue the command ContinuedFraction[Pi, 1000], replacing 1000 with whichever number one prefers. In this particular case, it only takes Mathematica a couple of seconds to give a thousand terms.