My own answers wouldn't be up to the subs standards, but while waiting for new answers you may want to check out what the FAQ has on it: https://www.reddit.com/r/AskHistorians/wiki/calendars#wiki_other_dating_systems

I can't answer the question as asked. But I do want to note that the question itself is not well framed. While much of the world uses BC/AD (more often now BCE/CE), it's not universal. There are several different calendars used on the Indian subcontinent, none of which have that BCE/CE break; there's the Jewish calendar which obviously never adopted such a distinction and in which 2016-2017 is the year 5777; it's the year 4715 in the Chinese calendar; in Iran it's the year 1396; and there are more.

Of course when crossing cultures, the others are (probably) universally normalized to our western calendar / dating system.

Looking at the faq, I see a lot of talk of various dating systems, but I don't see any discussion of critical dates, which is how I understand OPs question.

If i read the faq correctly, there seems to be 3 main groups of critical years: various regnal systems starting a count at a historic event like a crowning of a king or birth of popular prophet, the jewish system where they tried to estimate the date of actual creation by adding up the ages of the people mentioned in the bible, and a claim of cyclical calendars referencing the mayan calendar with a wiki link suggesting it has "a mythical starting point" at August 11, 3114 BCE.

My follow up question is: is that it? Are there any notable calendars which lack pivot years or use a novel system for them?

I'm having a hard time formulating a question regarding the mayan calendar, so I will probably just read up on it a bit more, but if someone could shed light on the type of events the mayans used their two counting methods for I would be much obliged.