Turns out that the SDL_TTF function TTF_RenderText_Solid is the culprit. My game over state basically said "keep on drawing the game over screen." Unfortunately, this included lines like this:
tempText = TTF_RenderText_Solid( font, "Final Score: ", textColor );
apply_surface( 350, 570, tempText, screen );
Every single call to TTF_RenderText_Solid was creating a brand new surface and assigning it to tempText. My faulty assumption was that the rendered surface only existed in the scope of assignment to tempText and then disappeared. This does not seem to be the case and it resulted in a heinous memory leak. For me, the solution was simply to have it render the text once instead of 30 times per second, but I could still see issues with regularly needing to update text on the screen (such as with scores or lives). I suppose you could pre-load a set of text surfaces (such as the single digit characters 0-9, etc), but that doesn't seem good either. I'm not sure if I'm missing something here or if I'm just running into limitations of SDL.
The good news is that I've been formulating some of the requirements for what Shmup02 will look like. There are still a couple gameplay bugs that I need to work out, but soon enough Shmup02 will be underway.