I have come across the following:
Person A: We made A LOT of money.
Person B: What's a lot of money out of curiosity?
I am wondering, does ending a question with "out of curiosity" sound natural?
"Out of curiosity" is a general phrase. It means you aren't asking the question for a special reason. You're simply curious. It's useless in your example -- the sentence would be better without it -- but it sounds natural enough.
A better example than "a lot of money" is asking a salesperson "out of curiosity, how much does that cost?". You're telling them you don't want to buy it -- you simply wonder about the price. Or "out of curiosity, what does your ferret eat?" means you aren't thinking about buying a ferret yourself. It's common in front of questions where asking them would make the person think you wanted to do something, but you don't. "How long do volunteers work, just out of curiosity?" means it sounds as if you might volunteer, but you don't want to. You're just curious.
If an accountant was calculating your taxes and asked "how much did your company make, out of curiosity?", that would be fine. It means they don't need to know the answer for their job.
The English stack exchange has an out-of-curiosity question (but I don't think the examples are as good).