Which theme is the most customizable?

Well, it all depends on your technical prowess. Two relatively friendly themes that offer a lot of flexibility are Thesis and Frugal. They’re not free but you get a lot of value for the price.

The default theme guarantees all features work properly. I have started with this and borrowed from other themes as needed.

Easiest though is to find a theme that comes the closest to what you want your site to look like and work. Take a good look at the templates to see if they are easily modifiable [some themes are so smartly coded that it takes a programmer to make even the simplest change], and start from there [make a copy, change the name in styles.css and work on that one].