Is it wrong that I want my girlfriend to pay for my travel costs?
Why would you make her pay for you?
She's rich and I want her to pay me back for all the times I've paid for her.
I get that you have helped her in the past, but you shouldn't pressure her to pay.  It's rude to expect your partner to pay for you to do things.
I doubt she will even mind though.
If you pressure your girlfriend to pay for your travel costs, she may begin to think you are using her or only with her for her money. It's best to pay for yourself and let her choose when she wants to spend her money on you.