Jasmines, Chinese Creeper (at least for a few years) come to mind.

I have a clematis in pot, it does all right. Obviously not as prolific as ones in the ground, but ok.

I take it you don't mean annuals like sweet peas and morning glory, which are both nice?

Good point Bryony - Sweet Pea and MG would be good climbing choices.

Climbing fuchsia, Lady Boothby will grow in a pot

Lady Boothby is not a climber though, just upright growing Steve.

Passiflora do pretty well in smallish pots. However it can leave the roots susceptible to hard frosts. Good for the more exotic species so that you can move them under cover for the winter.
Aserina scandens is an annual that grows well in pots too.
Gloriosa is ideally suited to container growing. Dry the pot out in early autumn and shove it in the cupboard under the stairs for the winter.
(Pasted from an earlier answer I gave elsewhere)

Also, Rhodochiton. It is tender though.

Black-Eyed Susan Vine (Thunbergia alata)? Just remember that any vine will only be healthy up to 4 times the size of the pot. I.e., a pot 1/2 meter in width and depth will support a vine 2 meters wide and tall.