Big numbers: can we really understand their meaning?

The problem with Eddington's number: Is it possible to count all the protons of the Universe?
I recently wrote an article about the 'unreality' of the Eddington number. The article title is: The Eddington number: a case for scientific arrogance?

Arthur Eddington (1882-1944) was an all-time advocate of the emerging Theory of Relativity since its introduction by Albert Einstein.

The 'Eddington number' is an extremely big number that supposedly represents the exact quantity of proton in the visible universe. This number, sometimes abbreviated as NEdd, needs 83 digits to describe it fully:

NEdd = 15 747 724 136 275 002 577 605 653 961 181 555 468 044 717 914 527 116 709 366 231 425 076 185 631 031 296.

The 'problem' with this number is that it is to difficult to 'digest'. How it is possible that with the incipient science methods we have, an the 'unelaborated' artifacts, instruments, and appliances we have to explore our surroundings, can anybody come to tell us the exact count of protons in the universe?

Read the full article for more info and rants about how other scientists and writers have dealt with the problem of big numbers.