Throwies with one LED and an old motherboard battery have been reported to last up to two weeks. I don't know how battery energy scales from tiny Li batteries to big alkaline ones (though wiki has some ideas), but I suspect they had more than enough power to keep them running 24x7 for a couple of weeks.
OK, same question. Would you rather have it run for 2 weeks or 4 weeks? (or 4 weeks or 8 weeks, etc...) Putting off the day you have to throw away the sign by doubling the life by adding a 35 cent photo cell is a good investment.