If bicentennial is the Latin word for the 200th anniversary, what word would one use for the 225th anniversary?
"Bicentennial" is not actually Latin; it's just English. It doesn't even come from a Latin word. In particular, bicentennial is an Americanism, and the more common word in England was (is still?) "bicentenary."
Presuming you don't want an actual Latin word, but an English word based on Latin roots, we can still make one up. "Bicentennial" comes from three chief Latin roots: bi-, meaning "two" or "twice," -cent- making "hundred," and "-enni-", from the Latin word annus meaning "year."
In Latin, a two-year period was called a "biennium," while a three-year period was known as a "triennium."
From these two arose a pattern in antiquity using the -enni- root, and later a period of ten years was coined using that pattern, yielding decennium.
Since English doesn't have proper case endings, we usually drop Latin's and adopt an adjectival ending, in this case -al, taken I presume on analogy from the Latin annualis, while that word literally means "one year old" in Latin, it was re-interpreted in English to mean "once a year."
From this pattern, modern writers coined the words "centennial" and, adding the bi- prefix, "bicentennial" (although in Latin "two hundred" is actually ducenti). The Latin for twenty-five is vigintiquinque, so you could theoretically have the monstrous bicentetvigintiquinquennial. I'm not personally a fan of this (or "bicentennial" for that matter), but I can imagine a technical type being attracted to it.
In the comments, Adam points out Semiquincentennial for 250th anniversary. The formation is instead "half" (semi-) "500" (quincent-). Perhaps quadrantinongentennial (one-fourth of 900 years) might be a more suitable formation. But as you can see from the variations listed in the Wikipedia article, there is no one way of saying this, and people are just making it up.
Dr. Wilfred Funk has passed your letter of July 11 on to us. We are happy to help, if you feel that you really want a new Latinate word meaning "one hundred and twenty-fifth anniversary"
The best model upon which to form the word is "sesquicentennial," meaning "one hundred and fiftieth anniversary." We need a first element meaning "plus a fourth," analogous with "sesqui" which means "plus a half".
"Sesqui" is apparently formed from "semis que," meaning "and a half." How, both "quarta" and "quadrans" mean "a fourth," so we may begin with either "quarta que" or "quadrans que."
The trick is to combine and shorten one of these as "sesqui" was combined and shortened from "semis que." If we follow the model of "sesqui" very closely, retaining the stressed vowel and final "s" of "quadrans," we get the word "quasquicentennial." Combining and shortening in other ways we can also get "quadqui-," "quansqui-," "quarsqui-," and perhaps several others.
On the grounds that it is closest to the model and also probably the least ugly of the set, I would choose "quasquicentennial" (pronounced kwahskwee-) as the new word.
So, naively, I think one could extend quasquicentennial to perhaps quasquibicentennial.
Of course, you may decide that you do not really want or need a new word. There is no point in proliferating them needlessly.
which is perhaps the best solution :)
I should add, perhaps, that this word would not appear in any of our dictionaries until it has established itself in wide currency, even if you should decide to use it.
This word now does appear in Merriam-Webster, though I don't see when it was actually added.
I hope we have been able to help you.
— Chapman, Robert L. (1965) "The history of Quasquicentennial" American Speech Vol. 40, No. 1
Looking in Google Books, however, maaaaybe some other people were using it in before, but that snippet revealed mentions the 125th of something started in 1841 = 1966. Most other results I get seem to be mis-dated, as they came out in the mid 60s.
Towards the "centennial" vs. "centenary" and "not everything needs a word", the definition of "centenary, centennial" in the 1926/7 version of Fowler's Dictionary of Modern English Usage (p. 72):
centenary, centennial, nn. meaning hundredth anniversary. Centenary, the usual British form, has the disadvantage that the notion of years is not, except by modern development, contained in it; this, however, is true also of century, and need not count for much. Centennial, chiefly used (as a noun) in America, has the disadvantage that it gives a less convenient pattern for forming the names of higher anniversaries on. As these are sometimes wanted, it is worth while to maintain centenary.
The shots made at these higher names often resulting in monstrosities [emph. added], a list of not intolerable forms is here offered. Bicentenary, which might have been ducenary, & tercentenary (trecenary), must be taken as established; but quatercentenary and quincentenary need not.