After DH and I had DS1 we both agreed that for a number of factors we would just stop at one child.
However, when DS was about 8/9 months I started to get feelings of wanting another. I didn't tell my DH at first because I wanted to see if the feelings would pass but when I was still wanting it and few months later I spoke to DH about it and he gave a very definite no. I didn't push it too much though as I knew it was me who had moved the goal posts.
As the months passed I did bring it up a few times but DH continued to say that no, he didn't want another one. I would put forward my reasons why we should and he'd put forward his reasons for why we shouldn't but we couldn't come to an agreement. We never actually argued over it but I did feel upset.
After about 12 months of me bringing it up and DH always saying no I out of nowhere had a bit of a breakdown about it and instead of talking about it light heartedly I told my DH seriously how much I wanted another child, why it was so important to me and how I just couldn't imagine never having another. It was the first time he'd seen me be upset about it or realised how much I wanted it and following that conversation he agreed we could TTC.
I fell pregnant straight away, we were both excited but unfortunately I miscarried about 4 weeks after my initial BFP. My DH wanted to start TTC again straight away but emotionally I just didn't feel up to it.
After another three months though I felt ready to TTC and DH said that was fine but if I had another miscarriage then we would stop trying. This upset me as I felt under a lot of pressure and so he then changed his trance to that we could TTC but if I wasn't pregnant by the time he was 35 then we would stop. That meant we had 12 months for me to get pregnant.
As the months passed I kept getting negative test after negative test and by the time I'd failed on cycle 7 I had given up hope that it would ever happen and I think DH had even forgotten we were actively TTC as we no longer really spoke about it. On cycle 10 however I got a BFP but it took me 4 days to actually tell DH as I was nervous about hiw he'd react. As it was he was shocked and it took a few days to sink in but then the happiness and excitement kicked in.
Our second baby is 4 weeks old now and DH admits it was the right decision to have another.
Does your husband know how desperately you want it?
What are your reasons for wanting another? I don't think just "wanting one" would be enough to make him change his mind. What benefits of having a 4th will there be to your family as a whole?
What reasons does he give for not wanting one? Are they practical ones?
With DH and I a lot of the conversations were based on me saying I wanted another baby and him saying he didn't and it wasn't until we actually spoke about our true reasons for our feelings could we start to see, understand and accept each other's view point. I was never angry or frustrated about DH saying no to another child because his reasons were perfectly valid.
If your DH has given you reasons for not having a 4th are they ones that you can understand or do you think they are just excuses?
I think to reach a conclusion you have to be really honest with each other and if DH really doesn't want another one then you will have to find a way to accept that.
You already have three children and I can imagine it would be very difficult to convince him to have a 4th, I found it hard enough to convince DH to go from 1 to 2.
I'm really hoping I don't ever have broody feelings again because I know there is absolutely no way I will get a 3rd child out of him 
Good luck OP 