The only annuals that will work well for more than a single season are those that reseed freely. A couple that come to mind quickly are alyssum and scrambled egg plant, Limnanthes douglasii. Forget-me-nots also reseed extremely freely, but get rather taller than what you may want as a groundcover and are also prone to powdery mildew.

Nasturtiums grow really easily from seed since the seeds are so big and make different size clumps depending on variety. They range from 9" to 3-4' in clumping ones and there are also trailing ones. I would think you would have to plant them again in another year. Godetia can be very good but sometimes it has not germinated well, maybe I didn't water enough. I would think you might want to start with what you could afford of a good groundcover so it could be filling in meanwhile.

Would the cost difference of perennial groundcovers be that different from annual groundcovers? Some of my favorite perennial groundcovers are creeping woolly thyme, creeping jenny (grows fast!), creeping phlox, blue star creeper, and vinca.

Dianthus grow well from seed if the area around the shrubs is sunny, and they make nice evergreen clumps too. Alyssum was my first thought as well for an annual, plus they smell good, particularly the white ones.

Wire vine is a perennial that grows very fast and is a very interesting shape. I would plant some perennials that you can take shoots off and start new plants for next year and intersperse them with your annuals.