Not a mathematician, by a long shot, but does this begs the question...Is something "wrong" with decimal? (Would a numbering system e.g. like base-pi/base-tau or base-e make any sense?)
However, we can have alternative, more powerful formal systems that are able to capture well-known transcendental numbers in a finite amount of notation. The most powerful class of such systems are Turing-complete systems. However, even such systems won't capture all transcendentals, since there exists a subset of transcendentals called uncomputable numbers. Uncomputable numbers outnumber computable ones.
More powerful formalisms also have the downsides that they lose some "nice" properties such as ability to always prove equality of two numbers, as they gain power. Algebraic numbers and especially rational numbers are super nice and well-behaved in comparison.
We start with the integers that, for the purposes of this comment, I will take to be god given. However, the integers are "missing" numbers in the sense that division of integers does not always give integers. For instance, the integers have 4/2 and 6/2, but are missing 5/2.
To accommodate this, we can invent the rational numbers, which extends the integers to provide closure under division [0].
As it turns out, the rationals are still missing numbers. For instance, there is no rational number that satisfies x^2 - 2 = 0, even though rational numbers can get arbitrarily close. To solve this, we introduced the algebraic numbers, which are the closure of the rational numbers under polynomial roots [1].
As it turns out, perhaps suprisingly, the algebraic numbers are still missing sum. For instance, the limit of the (1 + 1/n)^n as n -> infinity cannot be expressed as the root of a polynomial with rational coefficients. We define "transcendental" as the set of all numbers which are not algebraic.
You could easily define a base-e numbering system, which could, in principle, be useful for... something. A major downside is that you lose the ability to express most integers in a finite amount of digits, and doing arithmatic becomes highly non-trivial. Eg, we would have 1 + 2 = 10.0100112... (assuming I didn't make any mistake). It would probably be simpler to keep working in base-10, and instead explicitly talk about the coeficients of the infinite sum k_i * e^i
[0] Excluding division by 0. We could define something like the rationals such that this is defined. Actually we have and they are called wheels, but it generally creates more headaches than it solves.
[1] Eg, it contains all the roots of polynomials with rational coefficients. Technically speaking, the definition should say that it also contains the roots of all polynomials with algebraic coeficients, but it turns out we get this for free for having all the roots of polynomials with rational coefficients.
Most mathematicians are not concerned with which base the number system is in and are not really restricted in working in base 10. So in some sense almost all mathematicians don't really "use" decimal. In that regard, there are many possible interpretations to your question so probably many possible answers.
In other words, choosing to use base 10 instead of base 2 or 3 is like choosing to use python for your project because its easy, familiar and gets the job done rather than choosing a specialty language that would be "better" but harder to use.
[1]https://math.stackexchange.com/questions/446664/what-is-the-...