What are the "unimaginable numbers"?

The enormous importance assumed by the ancient Pythagorean School, and by Magna Graecia in general, for the development of mathematics and philosophy is well known. For example, the first true concept of mathematical-philosophical infinity appears in Magna Graecia (e.g., the Eleatic School) and this is also quite well known. A much less known thing, however, is that in the same times and places the idea of “unimaginable numbers” by the great mathematician Archimedes of Syracuse was born. In this paper we will explain what these numbers are and we will make a quick overview of the most modern notations used to write them and of some interesting (but unfortunately little disclosed) research topics that concern them.