Small companies or teams can sometimes struggle to provide career advancement for junior staff; it is not always that they don't want to help their teams advance, but that the "growth positions" are simply not there.
That said, I have worked for (mid-sized) firms where they understood they would have 50% turnover of new staff in the first two years (they told me this when I was hired!) and so invested in staff and provided opportunities accordingly.
Of course, this became a self fulfilling strategy over time (which I'm glad to say they broke out of, thanks to some inspiring new leadership after a merger.)
You sound like someone who is stimulated in part by change, growth and trying new things. If this is correct, then this need not being addressed will be at the root of your overall disatisfaction with your current role.
I would strongly suggest that you need to talk this over with your line manager or boss; running a small company is a busy process, and they may have just lost sight of your needs a little. Don't make this confrontational - over a coffee is fine - but let them know that you need a greater degree of career growth than you currently have, and your attitude towards your role is suffering as a result, and ask them for help.
If they have a "churn and burn" attitude towards staff you won't get much traction, but then at least you will know that you have to move on to both advance your career and to be happy.
If not, agree on a course of action - but always make sure that you have a project lined up to follow on immediately from any training you get as part of that action plan.
Finally - its worth noting that I tend to hire for attitude over technical skills, and I'm not alone in this. Someone who is enthusiastic, driven and wants to learn is more important to me than someone who can hit the ground running, but is closed to new skills or change.