Palms such as chamaerops are hardy here in Cornwall but most places I think. Slow growing and tropical. Yuccas look amazing but spiteful spines. Phoenix canariensis grows everywhere down here too but pretty hardy I think.
Phormiums in reds, pinks, yellows etc provide a tropical look but shrubs like pseudopanax, pittosporums and acacias add to it too. Olive trees in pots or in the ground? How about a red or pink cordyline?
The humble rosemary is a true Mediterranean plant with nice scented foliage and, if ??ou get the rich blue flowering ones, nice flowers too. Add lavenders and a convolvulous cneorum in a terracotta pot....velvety silvery foliage cascading with pure white flowers.
And cannas......large leaves in red, purple, green, variegated.....growing to 7' with fantastic exotic looking flowers in late summer. Grasses like stipa gigantea look exotic too....I currently have one 8' tall with plumes of golden flowers out shining everything else in the garden. Other grasses like miscanthus. I would avoid bamboo!
Maybe check out a hardy tropic nursery too....see what can grow for you.