How far does reliable counting of the years reach into the past:
- with the help of dating methods?
- using only primary sources?
Some astronomical events, viz. eclipses, can be reliably predicted to the day. I think they are the best method to identify a date exactly. This webpage provides some insight on the topic. It is basically a list of recorded solar eclipses, where the first one would be on 22 March 2134 BCE. There is however some uncertainty about whether or not the event really happened on the same day as the eclipse or was conveniently moved for political/religious reasons. I believe human calendars go as much back as eclipses, but this is based on calendars matching, which is subject to errors. The topics are, unsurprisingly, related. Furthermore, dendrochronology claims to be able to go back as far as 11000 BCE. It means that we can date the felling of that particular tree exactly to the year. However I am sure we can gather more expertise on this latter topic.
As for primary sources only, so far I have only identified the year 776 BC. The anchor event is the first (recorded) Olympiad, whereas the exactness of the counting is guaranteed by several sources recording the following Olympiads and using them for chronology, according to this guy-->.
: Chronology of the Ancient World, E. J. Bickerman, Cornell University Press, 1980
According to this paper, in 2951 BC there was a massive volcanic eruption. I do not know whether even more ancient eruptions can be calculated.