I don't have a problem with vegetables (except for mushrooms), so I haven't tried this with lettuce myself.

Buy a head of romaine, give it a fresh slice on the core end, put it in a wide mouth jar or vase with maybe an inch of (preferably filtered) water, and keep it on your counter. Pull off a leaf or two everyday. Give the bottom a fresh cut as needed and change the water every couple of days.

I keep my green onions sprouting this way. I imagine it'd work for romaine since it's long/tall in shape.