My son did exactly this, although fortunately for us the things he liked were 'healthy' in as much as I would usually give him some plain carbs with 3 or 4 different types of veg and some protein. I cook separately for him anyway, so this was no bother really, and as long as he was eating a balanced diet I wasn't too concerned. He ate the same 4 or 5 combinations for nearly two years, but at the age of 4 and a bit started to be much more open to the idea of trying new things.This went hand in hand with some kind of developmental leap whereby he was noticeably more confident, less concerned about staying in his 'comfort zone' and generally a bit more 'grown up' for want of a better term.
He has begun to eat a MUCH more varied diet based on gentle encouragement to 'practise school lunch'. We always talked about 'testing out' new foods so that he would know whether he liked it or not when he went to school, and we looked at the school menus online so that he could pick which elements he wanted to try first.
We also talked about how, when you try something new, your brain can be a bit surprised at the new taste, and it forgets to check whether you actually like the thing or not, so now we tell him to 'remind his brain' not to be too surprised and if he says he doesn't like something, we praise him for trying and say 'next time you try it your brain won't be surprised' etc.
This technique has been based on the fact that I was exactly like him and my parents went for the threats, the sitting at the table for hours and the endless discussion about food. As a result I dug my heels in, didn't expand my repertoire until I was in my 20s and have always had an unhealthy relationship with food.
When my son started down this road, I talked to my mum about the whole thing and what I was like as a child. I asked her if, to the best of her knowledge, her approach had ever actually worked. She couldn't recall a single time I had been 'forced' to try something and had actually liked it, and in fact she specifically remembers trying to make me eat certain foods that I still do not like to this day.
So, I go with what I think would have worked for me - we started out rewarding the trying of new foods, but he has now worked out that actually having more foods he likes to eat is a reward in itself, and he has only come across about three things that he genuinely doesn't like.
Sorry for rambling, but I am so pleased with the positive results we have had with minimal drama or conflict that I am a little evangelical about the whole process now. It may not work for anyone else, but as an adult who remembers being forced to eat faggots, I am possibly a little oversensitive about the whole food situation and didn't want my son to have the memories that I had of miserable meal times spent focussing on what I was or wasn't eating.