Hmmmm, how could you meet royalty?
Well, for starters, My Uncle John meets LOTS of royals, but that's because he works at a "ritzy" hotel over in New York. I mean, when Royals travel, they have to stay somewhere, haha. Problem with those encounters is, they aren't very personal, because most royals wish to be left alone and it can be hard to wade through body guards.
But that's only for the big-wigs. As far as royalty in general goes, you may have already met a few and didn't even know it! Some of the more distant royal relatives don't live under such tight rules, because they are not well known. Royals, just like us (of course), have aunts, uncles, cousins, half siblings, second and 3rd cousisns, nephiews, nieces, and all of those people have extended family too. These people don't really make a very big deal out of themselves, either.
I myself accidently bumped into one, and if it wasn't for someone telling me about it, I never would have known! My parents and I traveled to New York to visit with Uncle John and his wife. We were staying at their place. I came down to the hotel to ask uncle John when he gets off work (my mom wanted him to have dinner with us). I see my uncle through the windows, at the desk. He's on the phone and motions for me to wait outside for a minute.
As I'm standing there, I see a man comming out the front door, then stopping for a moment to make a call on his cell phone. He had an accent that I couldn't place. When he was finished, I called over to him, "Excuse me, do you know what time it is?" He glanced down at his watch and called back, "Almost 5:30." I thanked him and we stood there for a couple more minutes, talking (just casual chatter). Then Uncle John called me inside and the man went on his way.
Once I was inside, my uncle asks, "So how was your first encounter with royalty?" I stood there for a second, like...

"What??!" Then my uncle says, "Yeah, you know the King of *****?" (I can't remember the name of the country) I was like, "Uh....Not really, but go ahead." So my uncle says, "Well, that guy you were talking to is the son of the king's cousin." I was like, "Ok, well, I'm glad I didn't say anything stupid! Hehe." And that was the end of it.
The point is, such encounters are far more common than you think.