How many FIDE rated games are played per year? I was surprised not to be able to easily find this statistic. How many FIDE rated games are played around the world each year?
I was surprised not to be able to easily find this statistic.

The information is available with quite a lot of work from data which FIDE makes available on its website. The key page is the Ratings Download page. If you are an arbiter and you use a FIDE approved pairing program (e.g. Swiss Manager or Vega) then when you click the button to auto-magically update FIDE player details and ratings it will start by downloading and processing one of the files linked to on this page.
You don't have to be a pairing program to download files from this page. Anybody can do it. For me there are 4 important files -

Download full list of players (not rated included) ... XML format
Download STANDARD rating list ... XML format
Download RAPID rating list ... XML format
Download BLITZ rating list ... XML format

These files are updated by FIDE on an almost daily basis and so reflect data only for the last month. If you want historic data then towards the bottom of the page there is an option where you can select a period and see and download files from that period.
Most months I download the first file above and process it to extract these kinds of statistics in the same sort of way, I guess, as Kaspars Migla who runs the chessratings.top website referenced in the FIDE post in the other answer. If I forget then I download the 3 separate files from the archive. This is obviously more work but the big file isn't available in the archive.
From what Kaspars Migla says:

It is a very rough estimation because only games played by rated players in classical chess are taken into account.

it looks like he just downloads the standard file whereas I download the file with all data including rapid, blitz and Online Arena. So, his figures are likely to be slightly different from mine.
There are a number of things to note.

File 1 (full list of players) is more than sum of files 2, 3, and 4. It contains unrated players and ratings for FIDE's Online Arena.
For each player and rating type, along with the player's rating and K factor, it gives the number of rated games played that player in the period.
Because FIDE update the files more than once during the month you will get slightly different data if you download on different days.
Because these game numbers are for rated games played by a rated player they include each game played between two rated players twice, once for each player.
Similarly, they also include games played between a rated player and a previously unrated player who gains a rating in this period only once. The newly rated player has the game counted in their count, because it was played against a rated player, but the previously rated player doesn't because for them the game was not against a rated player. Hence odd numbers are possible in a sum which is roughly twice the number of rated games played.


How many FIDE rated games are played around the world each year?

OK, so now for the numbers. As I said I download and process the big file for all the players containing all the data rather than the three separate files for standard, rapid and blitz only. I've run some queries on my resulting database.
For each year the four numbers are sum(standard game count), sum(rapid game count), sum(blitz game count), (sum of sums)/2
As explained above the sum(standard game counts) is actually roughly twice the sum of actual FIDE rated games because each game is counted for each player.
As it happens the total number of FIDE rated games did exceed 3 million in 2019 for the first time although just over half were standard and the others were rapid and blitz.
I've given figures going back to 2011, the year before FIDE started rating rapid and blitz.
Year Standard   Rapid   Blitz   Total
2019 3015776    1684486 1338912 3019587
2018 2886482    1382951 1063591 2666512
2017 2786351    1150926 958753  2448015
2016 2629805    907642  785846  2161646
2015 2349963    631590  589853  1785703
2014 2022154    467333  426370  1457928
2013 1873662    313977  338094  1262866
2012 1689346    97722   117587  952327
2011 1569389    0   0   784694