It was pointed out to me that it seems like every year for the past few years has been a three day chag. So, I took a look at how common they are, and made some plots.
This shows the day of the week on which the first day of Rosh Hashanah occurs from 1980 to 2040. Rosh Hashanah can fall on Monday, Tuesday, Thursday, or Saturday. The three day chag years are those which fall on Thursday, highlighted in red:
Looking at this, it's clear that there have indeed been an unusually high number of three day chag years recently. From 2010-2014, all but one year were three day chags. However, a large gap is coming up -- from 2015-2023, there will be only one. Interestingly, we are in the middle of a long gap of weekend chag years, in which the first day of Rosh Hashanah is Saturday. The last one was in 2009, and we don't have another until 2020.
What about the other chagim?
The same question can be asked of the other chagim. The first day of Pesach can be Sunday, Tuesday, Thursday, or Saturday. That means that the first days of Pesach can form a three day chag from either direction, i.e. after Shabbat or before Shabbat. The last days of Pesach cannot form a three day chag. Here is a similar plot to the one above, although slightly different -- here I am showing which chagim occur as a three day chag, for all of the two day chagim of the year. Light blue indicates a Thursday/Friday chag, while dark blue indicates a Sunday/Monday chag: