Imagine a number which cannot be expressed as the ratio of two integers. You call it irrational because it has an infinite number of digits in its decimal representation, which never settles into an infinitely repeating pattern of digits. Yet any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the so-called infinite monkey theorem.
On March 14, 2019, Google announced that one of its employees, Emma Haruka Iwao, had found nearly 9 trillion new digits of that number, setting a new record. And less than a year later, on 29 January 2020, Timothy Mullican raised the bar to 50,000,000,000,000 digits, the most accurate value of your number so far. But abandon all hope. You would need all the energy in the universe to compute every single digit in its representation. A single number! (and people worried about blockchain 😉
Moreover, your number is not the solution (the root) of any non-constant polynomial equation with rational coefficients. You call it transcendental because it cannot be expressed using any finite combination of rational numbers and square roots or n-th roots, and It cannot be constructed with compass and straightedge. In fact, your number is the proof or the consequence of the proof that it is not possible to “square the circle”.
Squaring the circle was one of the great geometry problems of the classical antiquity. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. It took thousands of years, till 1882, to prove it was impossible. Today, only politicians persist in trying to square the circle, cause politicians do not understand about mathematics 😉
What do you do with a number which you cannot draw and you cannot even write down? You give it a name. You call it pi, you choose a nice graphical representation, π, and today you celebrate that you have language (and narrative) to hide behind a name a problem you will never be able to solve.