While transcendental numbers can be complex numbers too, let me assume that you implicitly mean real numbers here.
Every real number is an infimum of a set of rationals. Simply because the rational numbers are dense in $\Bbb R$, so given $x$, we immediatly have that $\{q\in\Bbb Q\mid x<q\}$ is a set of rational numbers whose infimum is $x$ itself.
Whether or not we can express in a [non trivial] formula in the language of mathematics (standard field operations, sigma notation, the "usual" functions that we accept to exist, etc.) what this set of rationals is a different question.
Some transcendental numbers will have to be referred by themselves, as the set above is defined in terms of $x$. It's not an issue, we already know that $x$ exists, so we don't need to define it from this set, rather we have all the real numbers, now given a real number $x$ we define a set of rational numbers that $x$ is its infimum. There's no circularity here.
Others we can express in other terms, e.g. $\pi$ can be approximated with certain sums and roots, and so we can easily define a set of rational numbers whose infimum is $\pi$. Essentially any number that you can constructively show to exist has a reasonable definition that allows us to write a [possibly very complicated] set of rational numbers that it is its infimum.