To the nth degree

‘To the nth degree’ began to be used to mean ‘to the utmost limit’ in the USA in the early 19th century; for example, this piece from The Petersburg Index, May 1871:

If there be any degrees of annihilation, Mr. Pengelly is fully titled to regard himself as demolished to the nth degree.

The term is borrowed from mathematics, where ‘nth degree’ equations and roots had been used for many years previously; for example, in The Monthly Review, 1794:

One of the roots of this equation of the nth degree is

The ‘n’ in the expression derives from the algebraic convention for an arbitrary integer (often an integer which may be arbitrarily large, that is, tending to infinity).