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Is it possible to represent $e=2.71ldots$ and $pi=3.14ldots$ without using the infinity sign nor using infinitely-many numbers, and algebraically, like representing the golden ratio, which is $1:dfrac{sqrt{5}+1}{2}$?

To summarize (some) of the comments:

The numbers that you can write algebraically, like the golden ratio--using any finite sequence of operations consisting of addition, subtraction, multiplication, division, square roots, cube roots, and/or higher-order roots applied to integers--is a subset of the algebraic numbers.

Transcendental numbers are real numbers that are not algebraic numbers.
It can be shown that $pi$ and $e$ are transcendental;
for example, see How hard is the proof of $pi$ or $e$ being transcendental?.

Hence neither $pi$ nor $e$ can be written algebraically.
So the answer is "no."

Yes. You can represent $e$ in one symbol "$e$" and you can represent $pi$ in one symbol by using the single symbol "$pi$".

However you can't express them as integers because.... they are not integers. You can't express $frac 35$ as an integer either.

You can't express them as a terminating decimal because they are not an integer scaled to a power of $10$.

Nor can you express them as a ratio between two finite integers because the are not rational.

Only difference between them and numbers such as $sqrt 2$ or the golden ratio which are also not rational, is that they are not roots to any polynomial with integer coefficients. $sqrt 2$ is the one of the roots to $x^2 -2 = 0$ and the golden ratio is solution to $x^2 - 2x -4= 0$. Neither $pi$ nor $e$ are a solution to any polynomials with integer coefficients.

Numbers that are solutions to polynomials are called algebraic numbers and the irrational algebraic numbers can't be expressed as finite decimals. But if we invent the notation, and we are mathematicians--- we can invent anything we like as like as it makes sense---, $sqrt[k]{m}$ to mean a $k$-th root of $m$, then we can express any algebraic number as some combination of roots.

Numbers, such as $e$ and $pi$ which are not algebraic are called transcendental. And, no, they can not be expressed via a digital decimal system. Nor can they be expressed as a series of root signs.

But so what? Using digital decimals to represent numbers is completely arbitrary and does not in any way make a number more "real" than any other. (The main reason we use digital decimals is because the are convergent and thus every thing can be approximate and accurately expressed through and infinite number of decimals if necessary.)

But to "represent" a number means nothing more than having a symbol for it. And if a number can by defined --- in a way that makes sense so that we know such a number exists--- we can define any symbol we want for them.

So I stand by my answer. You can represent $e$ and $pi$ with one symbol each: "$e$" and "$pi$"