Keep your eye on the ball
There are people in the world that will just never be satisfied. There is nothing you can say or do that will change this. No matter what you give them, they will want more. You just need to accept that. How are your other customers? If you only have one dissatisfied customer in a sea of 100s of happy ones, you're doing pretty well. You will never please everyone, you just need to please enough of them to stay in business.
That's the thing to remember: you're a business, not a charity. You don't need their gratitude, you just need their money. As long as they are paying you, you don't have a problem. You only have a problem if you think they are in danger of deciding they don't actually need your service after all, or taking their business to a competitor. Then you need to evaluate how far you are willing to go to keep this customer, and how that is going to affect the rest of your business. Is it worthwhile to retain this one customer if it means losing 5 others? Is the cost of implementing the features they want going to be profitable for you for what the customer is paying? It's all a balancing act, and only your company can know the right answer.
You will never please everyone. Focus on the grand scheme of what you need to do to keep your business growing.
Customer feedback is a good thing
You seem annoyed that the customer is requesting more features. I get where you're coming from. But the thing to keep in mind with every feature request, the customer is really telling you: "I use your product. I give you money for it. This is what I think will make it better."
You are under no obligation to meet every customer request. If it takes you away from your overall strategic vision, you shouldn't. But your customers are the ones paying for your service, and their requests are them telling you what will keep them coming back. You should be happy about this. The hardest part of any business is figuring out what you need to do to make money. Customer requests are them telling you exactly what they want so you don't have to guess.
Also remember, if these are things they want and you aren't providing, there is a good chance they will eventually start looking for someone else who will. Your goal is to make money. You can't do that if your customers start taking their business to competitors.
Strategic vision - the customer isn't always right
I currently work at mid-size SaaS company. Lord knows, we've lost customers, big ones, because we didn't provide the services they asked for. We also have prospective customers tell us all the time that they would consider our service if only we did this one other thing. But we decline, because we feel strongly that while they say those features would make that customer happy, it would actually hurt our business in the big picture. And it sucks. But we think that making those decisions actually put us in stronger position in the long run. We may have lost that one customer, but (we hope anyway) by declining their requests, we actually built a product that brought in even more business. And that's the main goal. (All I can say is we have a lot of customers angry at us right now, but even more that are throwing money at us and our business is thriving. The outlook is bright right now.)
Sometimes a customer wants something that would actually make your product less attractive to other customers. Sometimes it's something that would be nice, but will take your focus away from the bigger picture of what you are trying to do. You need to listen to them, but also evaluate it against what you are able to do, and what you are trying to do in the long term. Customers will tell you what they want, but you to decide if it's actually good for your business.
As the old Henry Ford quote goes, "If I had asked people what they wanted, they would have said faster horses."
Feedback is a gift. You don't want to stifle it. Your customers are telling you what they want so they keep giving you money. But you will never please everyone, no matter how hard you try. You need to evaluate their requests against what is actually good for your business and what is going to bring in more business. Remember: you're a business, you aren't looking for gratitude and good feelings. You're trying to make money.