infinite means without limit or end — extending indefinitely in space, time, or quantity, or so great as to be immeasurable — and, as a noun, names whatever has no bound.
The word entered English in the late fourteenth century from Old French infinit and Latin infinitus, "unbounded, unlimited," joining in-, "not," to finitus, "defined," from finis, "end." It is the plain negation of 📝finite, and the two words still travel as a pair. The noun sense, "that which is infinite," appears in the 1580s, and in theology "the Infinite" became a name for God.
In ordinary speech infinite is often hyperbole — infinite patience, infinite choices — meaning only very large. Mathematics uses it strictly: an infinite set has no last member, and an infinite quantity exceeds any assignable number. It differs from eternal, which concerns time alone and can also mean outside time altogether, and from endless, which stresses duration or repetition more than magnitude. Infinitesimal, its offspring, runs the other way, naming the vanishingly small.
📝James Carse gave the word a third life in 📝Finite and Infinite Games, where an 📝Infinite Game is one played not to win but to continue the 📝play, and whose rules change so that play can go on. In that usage infinite describes an orientation rather than a size.
I use infinite in Carse's sense. In 📝Bezos Won Capitalism I called capitalism a finite game and said we at 📝One Inc are playing an infinite one, played for the purpose of continuing play — the same word behind the 📝Capital Refractory.