After building a successful product, what do you do with the team when there is no work left and you can't think of something new?

Ask the team. What do they think is a good use of their time? No one likes to sit idle. If they were on a heavy schedule to meet the product launch, some downtime is useful to regenerate creativity, energy and focus. Have they documented all the processes used? Have you conducted a post-project review of what worked and what didn't and what could be improved next time?

These are some things for a temporary lull, however, the bigger question is how to engage your team when there isn't work to do. Can the team handle more projects?

Good luck!

Is the team on salary or paid an hourly rate for the work they're doing? Is the product successful such that it is generating revenue in excess of paying your team?

If you have the whole team is on salary, then I'd suggest you look at whether you can transition some of that team to contract work, working only when needed and for an hourly rate. There is a risk that that person might not be willing to move to contract and you might lose them, but if you're not actively utilizing someone, I don't see how you can justify keeping them on full-time.

I don't know how you're defining success but it's likely that you can build on the success you've already achieved. For example, there might be segments of users who aren't as engaged as others. Focus on increasing engagement for them. Or, focus on new in-app purchase options, etc., all focused on increasing what's already working to generate more "success" however you define it.

It seems like a somewhat unique situation you're describing. Happy to do a call to discuss in more detail.

That's a very unusual problem, I'll be honest I always have work.

My two cents,

If you have a solid team that has the ability to build a successful product do not loose them. Finding a team that can execute quickly and successfully will get you a long way and if you have one. DO NOT LOOSE THEM.

But I have to say, if the product is successful then in theory it should be growing like mad and that should pose multiple challenges on the infrastructure and the product itself. The team will have enough to do to keep the lights on else the product itself is not successful and needs something more to make it successful.

Finally, if the product is actually successful and is not growing due to external factors then

1. I would build a new/tweaked product that the market would accept
2. If you are confident about the product, I have put developers on support calls, user demos and shadowing sessions so that they experience what the users / customers are experiencing.
3. I like to have them as a fly on the wall in prospect conversations to give them context of what the market is saying ans what the needs are.
4. IF you have truly done all of the above and have a successful product, I would send the guys and gals on vacation and tell them to enjoy. They will be doubly productive when they come back and more appreciative than creating random unnecessary work for them.

But again DO NOT LOOSE THEM IF THE TEAM IS SO STELLER. (Actually can I hire them?)