My understanding is that bedding plants are annuals and flower for just one season.
I wonder if you are thinking of herbaceous perennials which may flower for several years?

Sounds like a simple question, but its not. First, the term 'bedding plants' is used quite specifically to refer to temporary, usually floriferous and colourful plants which are removed after 3 or 6 months, so things like busy lizzie, bedding begonias, pelargoniums. They're removed because they're generally not frost hardy.
What you mean is sub shrubs and perennials, herbaceous ones (which die down in winter) and simple perennials which don't die right back. These grow and flower again year after year.
As for which ones you might want, the conditions you want to grow them in are important - some are well suited to damp, shady conditions, others require full sun and well drained soil. Height and spread varies from 6 inches high by 24 inches wide to 3 feet high by 1 foot wide, so impossible to recommend anything without knowing where you're growing them. Also which part of the country you're in - some are more tender than others, so you'd use different plants in different parts of the country.

As bamboo said theyre not really bedding, but ones that you can leave alone and will hopefully come back everyyear are things like - echinacea, rudbeckia, coreopsis, heleniums. Or things like nigella and poppies self seed very easily.