Is it possible to add class features without multiclassing?  Our DM allowed us to take a free level in another class, but the level didn't count towards our overall level.  We basically received a few features from the other class as a bonus.  Is there anyway to do that in this system?  For example, I have a Druid that wanted to add some features, including spells, from a Cleric. 

No easily.

Only workaround is maybe making a subclass that replicates a subclass you want to take but also includes features for the level of the other class you want to take. This will be difficult and easily take quite a bit to get right depending on the class chosen, but it will be the best you will have.

Thanks, I'll give that a try.  

There are several feats that do this.

You can take Magic Initiate, which gives you 2 cantrips and once a day first level spell from a certain spell casting class.

There is a similar one that gives you a superiority die and several maneuvers from Battlemaster.

To be fair unless you're going to level 20 you could just add the level and remember, UNLESS you're using EXP