Depending upon your definition of the term “name”, the Googolplex (a one followed by a googol zeros) seems a likely contender. Of course, one can also construct compound names such as a “googolplex of googolplexes” to name arbitrary numbers as large as necessary.
This page may answer some of your questions about number names, or it may raise more questions. I can’t say, because I guess I really don’t understand the boundaries of your OP. Note that that author also claims a “googolplex” as the largest number with a name, but suggests other constructs as “googolplexplex” as possible.
As large as it seems, a googolplex is meaningfully expressible in exponential notation, which means that it’s a mere pittance. There are various special notations for very large integers, and it’s easy to come up with finite numbers that are beyond human comprehension. The problem is that it’s almost impossible to compare numbers in one notation with those in another.
I’m going to pick the Moser as the largest number that has a “name” in the sense of the OP. If you’re allowing any finite sequence of characters as a name, though, we should talk about the Busy Beaver function…
Well, it would be except that there are huge swaths of numbers in there that have no names.
The longest name that has a near-absolutely official name would be 999 vigintillion 999 novemdecillion 999 octodecillion 999 septemdecillion 999 sexdecillion 999 quintdecillion 999 quattuordecillion 999 tredecillion 999 duodecillion 999 decillion 999 nonillion 999 octillion 999 septillion 999 sextillion 999 quintillion 999 quadrillion 999 trillion 999 billion 999 million 999 thousand 999.
(You’ll pardon me if I don’t spell out the "nine hundred ninety nine"s, yes?)
I say “near-absolutely official” because once upon a time there was an alternative (chiefly British) naming convention that would render the number 1,000,000,000 as “one milliard” rather than “one billion”, and a thousand milliards was a billion. But that system has effectively fallen by the wayside. Milliards there may rarely be, but billiards are almost exclusively played on gaming tables and of trilliard and quadrilliards no one has heard at all.
I also say “near-absolutely” because there is a less official nomenclatural system extending upwards (mostly just by common prefix extension): a thousand vigintillions would be an unvigintillion, a thousand of those would be a duovigintillion. Makes sense enough, right?
After escating through uno, duo, tre, quattuor, quint, sex, septem, octo, and novemvigintillions, we would come (by extenion) to the trigintillions. An even thousand trigintillions would be an unotrigintillion, a thousand of those would be a duotrigintillion, and a mere 10 of those suckers would be not only 10 duotrigintillion but also your basic googol. (And now you see why “googol” isn’t incorporated into a normal numbering convention. It has a nice profound-sounding number of zeros after it, but those zeros don’t come in full packs of threes like millions, billions, and duotrigintillions).
Going on upwards,duotrigintillions would yield to trevigintillions once you got a thousand of them together, and 1000 of those = quattuortrigintillions…may I skip on, assuming youv’e got the pattern, and note that novemtrigintillions in the thousands are quadragintillions [123 zeros], followed (one full tier later) by qunquagintillions [153 zeros], sexagintillions [183 zeros], septuagintillions [213 zeros], octogintillions [243 zeros], nonagintillions [273 zeros], and finally centillions [303 zeros]?
And there I think you have it. I did not mention the googol plex because I did not come to it and pass it. That’s because I didn’t come to it, and shall not. I’m only up to 303 zeros. You can, if you wish, extend the naming convention using the same logic that got me to a centillion but you’ve got a ways to go before you get a googol zeros behind a 1.
And that prevents you from saying “999 googol plex, 999 <something>, 999 <something> <… very long elllipsis here…>, 999 thousand, 999” is the longest number. So many unnamed numbers, so little time
You’re the math dude, but as I understand the entry given, I thought it said that Graham’s number is the lowest-known upper bound for Ramsey’s number, which is somewhere between 11 and Graham’s number, inclusive.