There are quite a few year-numbering systems that identify years as being "this epoch" or "before this epoch", like AD and BC in English. And there are also systems like premodern East Asia, where a new epoch is announced whenever the ruling classes deem it necessary. (Edit: It turns out that they did this in Europe too with "regnal years" -- thanks suggestshistorybooks!) But are there any examples in history of societies with a system in between, where multiple epochs are recognized, but there are still a limited number and there's no standard way of starting a new one?
For example, imagine if we English speakers used "BAUC" for pre-753 BC, "AUC" for 753-1 BC, and "AD" for everything afterwards. That would be the sort of thing I'm thinking of.
Edit: This was a bit unclear, but I also want to exclude repeating cycles within epochs like days of the week, Metonic cycles, etc. I'm not after variant calendrical systems so much as actual epochs or "calendar eras" like BC and AD, BH and AH in Islam, etc.
I also want to restrict this to formal administrative contexts, so for example the informal "Xth century" naming system used in English doesn't count.
Edit: By "formal administrative contexts", I mean systems that were used by government officials (and their equivalent) to specify particular years. As far as I am aware European countries have never said things like "year 20, Century 16" in the way they do say "1620 AD". The centuries are recognized as subdivisions of the overarching epoch "AD" (or "BC" in some cases).
Medieval chronicles used a number of different ways to establish date. Besides anno domino, regnal years were frequently used -- referring to the number of years the ruler in question had been in power. This occurred most frequently with monarchs and popes, though bishops, dukes, and other highly ranked aristocrats were also used.
For example, the entry for the chronicle might read "In the 5th year of the reign of King John, in the year of our lord 1203..." or "In the 11th year of His Holiness Pope XXXXX, in the year of lord XXXX"
The Church also used things like the Metonic Cycle to help calculate important dates such as Easter (called Golden Numbers)
A medieval chronicle might also list the lunar day/month; Dominical letters to show the Sundays of any given year; and the Temporal and Sanctoral cycles - the former tracking the major events of the life of Christ and the latter tracking various Saints' days.
I'm not sure what you mean by "formal administrative contexts." "15th century CE" (XVe siècle in French; Siglo XV in Spanish; il Quattrocento in Italian; 15. Jahrhundert in German) is universally understood to indicate the hundred years between 1400 and 1500 CE. This format has been used in every history book published in the last 200 years I've ever read.
I think I answered the first part of your question, but I'm not positive. If you edit your question to be a little bit clearer, I may be able to help you a little bit more.