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Ad infinitum is a Latin phrase meaning "to infinity" or "forevermore".

Description

In context, it usually means "continue forever, without limit" and this can be used to describe a non-terminating process, a non-terminating repeating process, or a set of instructions to be repeated "forever," among other uses. It may also be used in a manner similar to the Latin phrase et cetera to denote written words or a concept that continues for a lengthy period beyond what is shown. Examples include:

"The sequence 1, 2, 3, ... continues ad infinitum."
"The perimeter of a fractal may be iteratively drawn ad infinitum."
The 17th-century writer Jonathan Swift incorporated the idea of self-similarity in the following lines from his satirical poem On Poetry: a Rhapsody (1733):

The vermin only teaze and pinch
Their foes superior by an inch.
So, naturalists observe, a flea
Has smaller fleas that on him prey;
And these have smaller still to bite 'em,
Thus every poet, in his kind,
Is bit by him that comes behind[1]

The mathematician Augustus De Morgan included similar lines in his rhyme Siphonaptera.[2]

Mathematical induction
Recursion
Self-reference
"The Song That Never Ends"
Turtles all the way down

References

"What is Ad Infinitum?". Kuknus. Retrieved Oct 27, 2020.

Swift, Jonathan (1733). On Poetry: a Rhapsody. Retrieved 14 December 2017.
De Morgan, Augustus (1915). Smith, David Eugene (ed.). A Budget of Paradoxes. II (2nd ed.). p. 191.