My all-time favorite I-would-die-if-it-ever-went-out-of-existence spiller is (say it with me) Creeping Jenny (also known as Lysimachia nummularia). Shown here, it's hardy to zone 3 so it comes back every year in my containers, it's a rapid grower and starts tumbling over the edges of my pots by mid-June. It seems to do well no matter where I put it on my deck whether it's in full sun or shaded by the leaves of my Japanese Maple. It takes all I have not to put it in every single one of my pots. But I have more suggestions and these should all be readily available:
Sweet Potato Vine (Ipomoea batatas)--some sophisticated gardeners would consider this a very pedestrian choice. That's just fine with me, I go with what works. I grow it every year and I love seeing the leaves spilling over the pots and stretching across my deck. By July, I am chopping these back every couple of weeks, that's how prolific they are. You can get different versions (colors) so there are options in terms of coordinating them with the colors of your other plants. Look for Marguerite, Sweet Caroline and Blackie. These are annual so don't look for them to come back next year.
Sweet Alyssum (Lobularia)--tiny flowers with a good spreading habit and it also reseeds. I will say that it doesn't perform the best in really hot weather but hang in there because when the dog days are over, it will bounce back and look great. Can usually be found in white, lavendar and pink.
Scaevola (or Fan Flower)--another annual, this baby loves the heat and seems to thrive on neglect, I love that in a plant. It grows 12 to 18 inches and cascades beautifully over your pot. I see them in white and in purple and invariably I buy the purple.
Thyme--just walk through the herb section of your nursery and you'll see all different kinds of thyme from English to lemon to coconut. Pick your favorite and plant it. It's a great spiller and the best part is, you can cut it and use it when you cook and the plant will continue to produce all season long.
So those are my "go-to" spillers. There are tons more including all kinds of ivy, although, be careful with those because they can sometimes be invasive. Also look for verbena, lamium and creeping wire vine. Just read the tag--usually it will say something like "makes a good trailing plant for hanging baskets and containers". If you see that wording and you like the plant (and you can provide the appropriate amount of sun/shade) then by all means, slap down that credit card.
Bookshelf: Container Gardening Books