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?
 A: 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.
A: 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
A: For $e$, the first few are $2, 3, 8/3, 11/4, 19/7, 87/32, 106/39$.  You get these by grinding out the continued fraction for $e$ and then computing the convergents.  See http://mathworld.wolfram.com/eContinuedFraction.html.
Also see http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfCALC.html where you can input a decimal approximation for the Euler constant and get the continued fraction and the convergents.
