A letter used to represent numbers is called a variable; the most common ones are 'x' and 'y'.
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That's called a variable.
A symbol, usually a letter, that is used to represent 1 or more numbers
"never-ending number" is ambiguous. You could either mean an infinite number, or an irrational number. For infinite numbers, you could use the Hebrew letter Aleph - which is used to represent the cardinal numbers of infinite sets of finite numbers. For irrational numbers, as far as I know there is no commonly accepted symbol to represent them.