A lot of this is paraphrased from the Google document I wrote these things out on. I'll be speaking of number size not on the levels of the largest googolisms, but in relation to more relatable numbers. Since I was tired when calculating many of these, feel free to re-check the math involved.
I started out with messing around with the googo- prefix, getting up to "googomdubar" until I got bored with it (I calculated it to be about 10^10^9.969, by the way). I then messed with "plex", to get a feel for it. ("googolbarplex" is 10^10^(2.5 * 10^5), by the way).
I then looked back at "googo-" and glanced at "googolple-" and decided, "I could use a similar vein of thinking and make something a tad bit stronger" and came up with "-ple-", which is technically an infix. Like "googo-", a roman numeral is placed on the end; if the roman numeral is n, then "-plen" is n^y, where "y" is the number at the fromt of "-ple-", such as googol. It's not especially powerful, per se, but it opens the way for somewhat stronger "-duple-", "-triple-" and so on - "-plemdubar", for instance, is 10^9y. "Googomdubarplemdubar" does not make it to a googolduplex, but is 10^10^10^9.949, which is by no means small. By comparison to -plemdubar, -duplem is 10^10^(3y + log3) (or 1000^1000^y), and -triplem is 10^10^(3y + 2log3) (or 1000^1000^1000^y). -centiplem would be 10^10^(3y + 99log3) or (1000^^100)^y.
I also expanded on "googo-" and made what's essentially a function, but in word form. Basing it off of "googol", the first letter(s) represent the number base, by taking the numerical equivalent and adding three. This itself is base 26, so "aa" is 27+3 or 30. The number base simply changes what the powers of ten are equivalent to; "boogol" is 10 base 5 to the power of 100 base 5, or 5^25.
The first set of vowels represent the power of the (hyper)exponent; two means that the exponent is 10^2.
The middle amount of "g"s is likely the most powerful part of the entire thing; it represents the amount of up-arrows. For example, googgol is 10^^100, a power tower of 10's 100 members high. Not bad.
The second set of vowels represents the power of the base. It's good to note here that I came up with a way to shorten the two sets of vowels; where a single o makes 10, 5 o's can be shortened to one u, 5 u's to 1 a, 5 a's to 1 e, and 5 e's to 1 i. A single i is equivalent to 5^4 o's; a gigol by this system is 10^10^625. While drastically smaller than Bowers' gigol, it still is reasonably big, and does outclass a googolplex. The contracting-into-another-vowel system can either be continued into special character vowels in unicode order, where 1 a-grave is 5^5 o's. The faster way is allowing the vowels to form exponents, although this causes pronouncibility to take a severe hit. That's an issue that may need to be resolved eventually.
But with allowing the vowels to form exponents, you get i^u, which is 625^5 o's, or 5^20 o's. After this, there's i^a, i^e, and i^i, which are 5^100, 5^500, and 5^2500 o's, respectively. This makes a gi^igol 10^10^(5^2500). Impressive. But exponents can be stacked, giving i^i^u, i^i^a, i^i^e, and i^i^i, which are 5^(4 * 5^20), 5^(4 * 5^100), 5^(4 * 5^500), and 5^(4 * 5^2500) o's, respectively. A gi^i^igol is really big; 10^10^(5^(4 * 5^2500)) is by no means small!
But let's stack the stacks. i^i^i^i o's is 5^(4 * 5^2504). Because exponents aren't always intuitive, this change is actually approximately the same as multiplying the amount of o's by 10^10^3 (actually closer to 10^10^2.8)! In fact, every extra i added to the power tower does the same. Knowing this makes resolving earlier tetrational values somewhat easier.
i^^a is 5^(4 * 5^2588). i^^e is 5^(4 * 5^2988). i^^i is 5^(4 * 5^4988). And this is just the amount of o's involved! Obviously a gi^^igol is huge!! (it's 10^10^(5^(4 * 5^4988)), so yes, yes it is).
How about... stacked exponents in the tetra-exponent? (i^^i)^i has 5^2500 members! 4 * 5^2500 dwarfs 4988 by so much that ignoring that completely still gives a very accurate approximation, although the real number is slightly bigger (relatively). (i^^i)^i is actually 5(4 * 5^(4 * 5^2500)). Evaluating it in terms of powers of ten is beginning to push the limits of big number calculators, although the number itself has far, far exceeded the largest limits. Every "i" added to the power tower in the tetra-exponent nests another (4 * 5^2500), as demonstrated by (i^^i)^i^i, which is 5(4 * 5^(4 * 5^(4 * 5^2500))).
Stacked tetration pushes the limits of what I currently can practically do; i^^i^^i far, far, FAR dwarfs (i^^i)^i^i. My calculations give it as a power tower 5^(4 * 5^4988) members tall, consisting entirely of i's! And remember, an i is equal to 625 o's. Don't get me started on i^^i^^i^^i. If I am not mistaken (which I totally may be at this point, my element is power towers, not tetra-towers), is a power tower i^^i^^i members tall.
And all of this can be hurled into the vowel portions of the "googol" thing I came up with well up on the page. Granted, when dealing with tetra-towers, the effect of taking 10 to the power of it isn't as drastic as intuition may make you think, but using the most powerful operator, the middle operator, and it can all work to scale.
Oh, and the last letter is a roman numeral and is there for retroactive definition, because I didn't want to get rid of that yet.
Since the amount of g's in the middle is the most powerful part, I'll work on a way to condense them and permit larger and larger hyper-operators.
I'll likely make a Google webpage for this, so it remains more organized.