Plenty. I talk with patients for a few minutes during a very stressful and scary time in their lives, reassure them, get them off to sleep, wake them up later with a few words, and typically never speak to them again. It's perfect.
Sometimes there is pain service work or postop followups to do with inpatients. Short rewarding encounters. Close to perfect.
For regional cases, especially c-sections, there's more interaction during the case, but it ends and then they're gone. Edging a little further away from perfect, but still pretty awesome.
Here's what I never do - see a patient in clinic, make an incremental change to a BP medicine, and see him 3-6 months later for followup to discuss why he didn't fill the new prescription. This state of non-affairs is gloriously perfect.
Short and meaningful patient interactions (that end, soon) is one of the greatest things about anesthesia. For people who like the idea of establishing long term relationships with patients, maybe being the classic small town doc that delivers a kid and 25 years later delivers that kid's kid ... anesthesia isn't a great fit. There's some potential with a chronic pain practice, but that's not for me. It might be for you.