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to infinity; endlessly; without limit.
Contemporary Examples

Good versus evil, light versus dark, a snake eats its own tail, ad infinitum.
‘Hemlock Grove’: Netflix’s Latest Original Show Is Scary Bad Jace Lacob April 18, 2013

Historical Examples

As the atoms are eternal and uncaused, so is motion; it has its origin in a preceding motion, and so on ad infinitum.
Encyclopaedia Britannica, 11th Edition, Volume 8, Slice 2 Various

There have been republics (so-called) ad infinitum and ad nauseam.
The Arena Various

So the frontier dispute dragged on ad infinitum till the British Consul left it to settle itself, and went back to Tabriz.
The Cradle of Mankind W.A. Wigram

Every mechanical cause has its cause in turn, and so ad infinitum.
A Critical History of Greek Philosophy W. T. Stace

Some have single bases, a round on a square, or vice vers, and so on ad infinitum.
The Cathedral Builders Leader Scott

And so on, ad infinitum; it is enough to give the hint for the play.
Froebel’s Gifts Kate Douglas Wiggin

She would arouse at first a very few, and then a few more, and a few more, and so on ad infinitum.
Stories by American Authors, Volume 6 Various

Fall in, and you are expected to stay in, ad infinitum, ad nauseam.
The Wit and Humor of America, Volume IV. (of X.) Various

But why could we not also suppose a third duty, commanding us to observe the former, and so ad infinitum?
Elements of Morals Paul Janet

adverb
without end; endlessly; to infinity ad inf

Latin, literally “to infinity” from infinitum “infinity,” neuter of adjective infinitus “endless” (see infinite).