Well depending on how much of your driveway, you could build a moveable box(es?) on wheels. In my case, my husband would never go for it! Actually, until we had about 17 HUGE trees cut down, the driveway was our sunniest spot too! I just didn't get to garden much!
As far as watering, I sometimes hook two hoses together, other times I will lug my watering cans to all the parts of the yard that the hose doesn't reach. It really depends on the weather! If it's 95 I don't want to lug the can I drag the other hose out. You just need to think about it all before planning and building. I think if you had boxes on wheels you could move them out of the driveway if guests were coming, etc. (Several times a year we need our entire driveway for cars and wouldn't want anyone running over my gardens! LOL!)

I tried a container garden on my driveway this summer, because my regular veggie garden was out of commission. When we had this record heat wave, I had to hand water everything twice a day to keep my plants alive. As a result, the tomatoes have blossom end rot and look pretty wilted, but they're still alive. Maybe you'd have more luck with a SFG because the beds, being larger than containers, wouldn't evaporate as quickly, but a simple drip system on a timer might be good insurance against fried plants.

I have raised beds along the side of my driveway. I live below the hill, and have a steep bank in where my driveway is. I just put them there, permanently.

I was looking back over this post and wanted to update. I finally decided to splurge on a self-watering 3'x 3' planter from Gardener's Supply. It wasn't cheap, but I have had great luck with it. It is about 12 inches deep, is very attractive, and is on wheels. It has been very productive over the last two years. I have grown tomatoes, peppers, lettuce, beans, cucumbers, brocolli, etc. The tomatoes didn't taste great,potentially due to the relatively bland potting soil, so they are back in the ground, but I am so happy about this method. The lack of dirt surrounding the planter seems to keep pests down, the black driveway seems to help radiate heat, it is a decent size, and it can be moved to take advantage of the sun. I would NOT try this without the self-watering aspect of the container as it would dry out really fast, but overall, I have been very pleased. If I have time and the knowledge, I wil try and post a pic soon.

TBT