Getting back to the broader philosophical issue, separated from "what pizzas do you like?" there is a significant variety of pizza style in Louisville.
Certainly (as reported in Food & Dining a while back) the most characteristically "Louisville pizza" is the thin crust, piled high, cheese plus toppings plus more cheese mountain best exemplified by Impellizzeri, the Big Wick, the top-end Clifton pie and a few others. Most other local pies (Bearno's, Za's, Jockamo's, etc.,) are generally somewhat reduced versions of this. I'd put RichO's and Charlestown in this category too, with the footnote that they aren't piled high like the originals but rank among the tops for quality and flavor.
Windy City, as mentioned, has the city's best deep dish Chicago style pie, but it should be noted that they do an excellent thin-crust model, too.
I love Boombozz but would put him more in the category of "modern creative" pie, even though you can certainly get a very good standard thin-crust pie there.
I agree on Hero's as the top NYC-style, although Luigi's certainly replicates a NYC street-corner by-the-slice shop, for better and for worse.
And even though I grew up a Kentuckian, not a Hoosier, I have to admit that I can embrace the guilty pleasure of the "party-cut" Midwest/Hoosier thin pie with the toppings all the way to the edge, a la Pizza King and Uncle Tubby's.
Lots and lots of options, but if you want to show them the one style that seems particularly linked to Louisville food culture, I'd go with Impellizzeri's on Bardstown or, in second place, the Big Wick.
