I have a friend who has exhibited Jeanne LaJoie, but it definitely must be sprayed for blackspot.

Do they have to be climbers?

Diane, I asked about climbers because I thought they would have larger amounts of blooms and longer stems to use in arrangements with other types of flowers. Thanks for the info.

I don't think most climbers will do what you want. The long canes
usually have shorter offshoots with the blooms. Try some of the taller regular bushes; check their descriptions in the catalogs.

Many minis grow to 3'+ tall, as well as mini-floras, and can get really long stems. Conundrum (a mini-flora) is taller than Miami Moon (a Floribunda). So, you don't have to limit yourself.