Interesting question indeed.
Two dramas I've started and dropped multiple times have been mentioned here: The Greatest Love and City Hall. I have a strong feeling I could love both, if only I could get past the male lead - which, right now, I can't. So, although something major happens to my taste buds, I fear I'll never pick them up again.
There is a drama though that I had dropped by the first episode and later became one of my favourites: Stand Up!!, j-drama, 2003. Possibly, I was a little too inexperienced in the Asian drama world, and thought the plot and the overall acting was ridiculous. I'm glad I picked it up again several months later, because I loved it and tried with all my might to make my man watch it too - which resulted in the nth failure. Oh well, I can't have it all, can I? :D