What could be the possible downsides of sticking with an old version of OSX? I've bought a Macbook Pro a pair of years ago, and it came with OSX 10.9.5 (Mavericks). Since I am very happy with this version of OSX, I've ignored all the updates since then, and I'm still running OSX 10.9.5. 

I'm afraid that upgrading OSX would leave me unsatisfied, so basically I'm sticking with a version I like (and by the way, I also prefer this version from an esthetic point of view!).

However, I'm wondering if there could be some problems when never upgrading OSX. What are the problems that could emerge in the long run if the OS is not upgraded?
there MAY be a small risk of issue long term with staying back on OS levels. However that issue is pretty minimal.  You should make sure 10.9 is updated to the latest flavor, to prevent any major issues. (I happen to know that 10.9.5 isn't missing anything from 10.10 or 10.11).

The tradeoff is that an older OS will also (GENERALLY SPEAKING) work better on a machine that's not brand new. 

Personally I own 100+ Macs, and run all my personal systems on 10.9.5... I hate 10.10 and 10.11 - both are filled with issues, both slow down computers.

My recommendation would be to stick with the older OS as long as you can, understanding that you'll eventually run into issues with add-on's (Flash, youtube, etc)... The core OS will run just perfectly forever (I've got a couple in-house servers running 10.5!), but you'll want to eventually upgrade when some of these ancillary products give you problems....