I actually heard this expression a long time ago, didn't really pay attention to it, and honestly forgot all about it, until now.
And like most words and phrases in Spanish, it can have different meanings. The first meaning is "right there". Let's take a look at some examples.
Cuando entras la puerta, está luego luego.
When you walk in the door, it's right there.
El catálogo de pasteles está luego luego a la entrada, lo vas a ver.
The cake catalog is right there at the entrance, you'll see it.
It also carries the meaning of "right away"
Disculpe, me gustaría un vaso de agua.
Claro, se lo traigo luego luego.
Excuse me, I'd like a glass of water.
Of course, I'll bring it right away.
Another way to say "right away" is "en seguida", which is something all Spanish speakers will understand. Luego luego is something that you'll most likely only hear with Mexicans, or might carry a different meaning altogether.
There's a great thread in WordReference that talks about this phrase, and also points out that it means something a little in Spain. There's also some more great examples. Here's the link:
Well, that's it for luego luego. But have no fear, I'm going to get started on my next entry luego luego.