I have a lot of friends that do it, only do it a little bit myself.
Pretty much the only thing I make myself in this fashion is an apple-ginger smoothie with a heavy influx of greens (spinach is the mildest; kale is a bit harsher, but palatable). A splash of apple juice for sweetness (4-8 ounces), a few small chunks of fresh ginger (peeled), and a couple hearty handfuls of spinach or kale. That's a good way to get a big influx of greens in you, the ginger has a nice bite, and the apple juice is just sweet enough to make it drinkable. Only do this if you can afford a splash of sugar every so often in your diet, as even a small glass of apple juice is pretty high in sugar.
Several friends from my yoga studio (quite a different demographic from the typical T-Nation member lol) make and sell their juice at the yoga studios (juicing and vegetarianism are both big in the yoga community around here). I have sampled their wares frequently and come away impressed. Combinations that I have fond memories of:
Carrot apple ginger
Cucumber watermelon mint
Beet...shoot, I can't remember, but they make one with beets and some other stuff that's really good (I want to see it also includes apples, but I'm not sure)
Hopefully that will give you a couple of ideas.
Broccoli: terrible idea. Stick to vegetables that have a fairly mild or sweet taste to them. Carrots and beets are both winners. Spinach greens are pretty neutral if you have something else like apple or ginger to give it flavor.