I assume you're confident with the care you've had. While it is safe to stay long term on anti thyroid drugs, it's understandable that you want stability while on them.
Have you been on and off? On consistently? What dose? Do you split or take once a day? Has your doctor offered you the option of block and replace therapy? It's quite effective when administered properly.
There is a Graves support group on yahoo that I haven't checked in on in years but that I found very helpful. Also, Elaine Moore's work.
P.S. about bad effects on the liver: that's why they do a metabolic panel. There are many medications that can be problematic, such as immunosuppressants given for Crohn's. GI's allow a patient's level to be up to 5 x the norm before stopping.
The difference between Crohn's meds and ATDs is that the former are started low and one builds up to a long term therapeutic dose. In the case of the latter, one starts high and lowers the dose, so you can see that there is less stress on the liver. You might want to look into milk thistle (get reliable info) of a diet that's easier on your liver, e.g. less animal protein.
Its been on consistently for 3 years. I was on methimazole since my diagnosis and regulated my dosage accordingly to keep my hormones at an average level. Its just recently that my levels went up and it seems reasonable now to finally end the medication path and take the RIA uptake instead.
As for other options like block and replace therapy, I have no idea what that is and wasnt offered by my endocrinologist. Could you enlighten me? Thanks