Can We Represent Every Real Number Using Only Finite Memory?

There are in fact indefinable real numbers. If you have a language with a countable number of symbols, then any formula $P(x)$ defining a real number will have, at best, a countable number of symbols. By a Cantor-style diagonal argument, you can only define a countable number of reals.

To elaborate: working in standard first-order logic with a given set theory, you can call $r$, a real number, to be definable if there is a formula $P(x)$ such that $r$ is the only real number such that $P(r)$ is true. However, note that the collection of formulas with one free variable is countable, whereas the collection of reals is uncountable; hence there are uncountably many undefinable real numbers.

For more information, see this wiki page; Timothy Gowers also has a good expository article that may be of interest.

(So the answer is no, you can't represent every real number in finite memory, so to speak, if you assume that every description can be represented as a formula in first order logic.)

So do any reals that cannot be described/represented in finite memory exist? For which their only closed-form expression requires infinitely many digits?

Yes, the fact that transcendentals are uncountable implies that there is no finite or even countably infinite way to represent them all. ( Assuming memory is discrete and therefor countable).

Further more, there are numbers that can not be computed, so how does one stores an uncomputable number? ( like the Chaitin's $\Omega$ constant)