It all depends on what you want to grow. As Brooke said, lettuce would be fine - as would pretty much any salad leaves. You could also try spinach, herbs and trailing strawberries.
If you've got room to transplant, starting pretty much anything off in guttering before moving it on to the ground is a great idea. It allows you to constantly have things waiting in the wings for when other crops in your garden are used up.
Just make sure to put drainage holes into the guttering and keep the soil well watered and you should be fine - it's worth experimenting.
There's a useful site showing you how to do it here: http://lifeonthebalcony.com/keep-your-mind-out-of-the-gutter-but-its-ok-to-garden-in-one/
Good luck :-)

radishes are great for that matter, herbs too, but also little colored beets and some short carrots can be sow in such narrow and not so deep planters. the trick is to choose dwarf.short varieties that will be suitable to your climate and exposition.
the watering is also the key, try to build a system of drop by drop watering, it will help a lot to maintain the crops healthy.

I would also advise to chose the rain gutter in the largest and deepest model possible, just 2inches more deep could radically change the harvest :) .