Is it possible to represent every huge number in abbreviated form? Consider the following expression. 

$1631310734315390891207403279946696528907777175176794464896666909137684785971138$
$2649033004075188224$
This is a $98$ decimal digit number. 
This can be represented as $424^{37}$ which has just 5 digits.

or consider this number:

$1690735149233357049107817709433863585132662626950821430178417760728661153414605$
$2484795771275896190661372675631981127803129649521785142469703500691286617000071$
$8058938908895318046488014239482587502405094704563355293891175819575253800433524$
$5277559791129790156439596789351751130805731546751249418933225268643093524912185$
$5914917866181252548011072665616976069886958296149475308550144566145651839224313$
$3318400757678300223742779393224526956540729201436933362390428757552466287676706$
$382965998179063631507434507871764226500558776264$

This $200$ decimal digit number can be simply expressed as $\log_e 56$ when we discard first $6$ numbers and then consider first $200$ digits.

Now the question is, is it possible to represent any and every huge random number using very few characters as possible, theoretically.

...Also, is there any standard way to reduce it mathematically?
No.  The problem is very simple: there are more huge numbers than abbreviated forms.  Suppose that I allow you to use the numerals 0-9, the English alphabet, and spaces to describe whatever numbers you want; that's still only $37$ characters. There are only $37^{140}$ expressions you can write down using these characters which is $140$ characters or shorter (the length of a tweet!), so there are only $37^{140}$ numbers that you could conceivably tweet to someone else.  In particular, since $\log_{10} 37^{140} = 219.5$, this implies that there is at least one number with $220$ digits which is impossible to tweet to someone else without using something other than numerals and English.  

All right, I'll be even more generous: I'll let you use all $127$ ASCII characters. That means there are $127^{140}$ possible tweets, some of which mathematically describe some kind of number, and since $\log_{10} 127^{140} = 294.5$, there is at least one number with $295$ digits which is impossible to tweet to someone else (again, assuming that Twitter uses ASCII).  I could give you the entire Unicode alphabet and you still won't be able to tweet, for example, the kind of prime numbers that public key cryptosystems use.

These kinds of questions are studied in computer science and information theory; you will find the word Kolmogorov complexity tossed around, as well as the word entropy.
The way I understand this question, the answer is "yes." Simply write the number in, say, hexadecimal notation, and it will always be shorter than in decimal (if it is >=100000). If you are looking for the shortest representation, however, that is well defined only if you specify exactly what counts as a valid representation of a number. After all, every number you come up with can technically be represented in just one digit if you choose a sufficiently large base (at least one higher than the number itself).

About the standard way to reduce it: I'd say the decimal system is the standard way of reducing numbers from just writing the appropriate number of dashes. :-) In the sense pointed out by Bill Dubuque, there is no representation that is more space efficient in principle, that is if you transfer the problem to a more abstract domain. If, instead, you are looking for the shortest representation using a given set of mathematical symbols, then no, there is no standard way to find it. I also suspect no algorithm can do better than brute force search (that is, first check all numbers representable using 1 character, then those representable in 2 characters, and so on). In any case, nobody would write a number as "loge 56 (log 56 base e) when we discard first 6 numbers and then consider first 200 digits," so no conceivable set of standard symbols will account for that anyway.