Does every irrational number have continued fraction representation? Are CFs fastest method so far? (Rational approximations of $ e,$ and $\gamma$) What are rational approximations to $e$ and Euler-Mascheroni constant $\gamma$ ?

(Like $ 22/7 , 355/113, ...$ are for  $\pi.$) 

EDIT 1: 

(Number theory is not my cup of tea..). I wish to be able to learn if for every irrational number there exists an infinite set of progressively reducing 
error rational numbers as approximation and a procedure by which it is obtained.  

EDIT 2:

Are continued fraction the fastest or the only method to obtain rational successive  approximations to irrationals?
You can calculate continued fraction representations, where you can choose to terminate after you've reached some level of accuracy: 

$$e=[2;1,2,1,1,4,1,1,6,1,\dots]$$

Here are some of the first results: 

$$e\approx\frac{19}{7}, \frac{87}{32}, \frac{106}{39 },\dots$$

Another (beautiful, but non-simple) continued fraction representation of $e$ is $$e=2+\frac{1}{1+\frac{1}{2+\frac{2}{3+\frac{3}{3+\cdots}}}}.$$



Same for $\gamma$:

$$\gamma=[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40,\dots],$$

where the first few (useful) convergents are 
$$\gamma \approx \frac{11}{19}, \frac{15}{26},\frac{71}{123},\dots$$



Edit as a response to OPs edit:

From the Wikipedia-page: every irrational number $\alpha$ is the value of a unique infinite continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values $\alpha$  and 1.
Euler's approximation (with function in link):
19/7, 87/32, 106/39, etc...
http://www.johndcook.com/blog/2013/01/30/rational-approximations-to-e/

Euler-Mascheroni constants can be found here with references to the original findings:
http://mathworld.wolfram.com/Euler-MascheroniConstantApproximations.html