If I wanted long term patients, should I consider a different field? I suppose I wouldn't mind even if it was a smaller percentage of my patients as long as there are some. Certain fellowships, besides H&N?
Medicine mainly treats, rather than cures disease. There are a few exceptions, including antibiotics (not that exciting), or chemotherapy for certain types of leukemia/lymphoma. But usually you are talking about regular follow up visits to review the patient's medicine list and make sure they are still on the appropriate regiment.
Meniere's is similar to medicine. You start with a medical treatment algorithm and progress through various procedures and surgeries, with symptom improvement considered success, not necessarily cure of disease.
In reality, it is hard to rank specialties on how well you establish long term relationships. I would consider ENT to be pretty good for that, but if you are looking at a 2.5 visit per patient number you might not think so. ENT (and most fields, really) have a lot of patients that need nothing done, except maybe a referral to another doctor. For example, someone who thinks they have Meniere's, but really needs a cardiovascular workup. You may be indicated to get an imaging study or other testing to reassure the patient that their symptom is not related to anything serious or treatable (by you). So, this group of patients, which I feel is pretty large - I would propose at least 10 to 20% - has only 1 or 2 visits. I think this type of scenario is common with specialties. I would say most otolaryngology practices have some patients that come back long term, but that is not a majority.
So, if you are really serious about having a good long term relationship with your patient, I mean you are prioritizing this very highly, then consider primary care. You could treat general medical problems, refer you patient to consultants as needed, get reports back from the consultants, and stay involved with every aspect of a patients care over a long time period. With family medicine, you would be involved from infancy to end of life.
Or, it may be that you think it sounds neat to see patients long term, but you are just considering your options... In that case, take some more time to consider various fields of medicine and surgery, and consider other aspects that are good and bad rather than just having a good long term relationship.