Standby fares are still a thing in the land of the future that time passed by: Japan! In particular, Solaseed Air offers a Visit Japan fare where any flight is just ¥10500 if you’re willing to show up at the airport and take your chances:
http://www.solaseedair.jp/eng/fare/price/visit.html
Now I’m not entirely sure why you’d want to use this, given that both JAL and ANA offer regular “book and fully confirm in advance” tickets on all flights for ¥10800 with their respective Visit Japan fares, but hey, you did say anywhere…
Most airlines eliminate cheaper seats as you get closer to departure. The vast majority of last minute bookings are business travelers who will pay whatever it takes to get there when they need to. So no real motivation to sell discounted seats (especially since last minute for business needs can truly be last minute).
The route you are looking, USA to southern Europe, will be busy in spring, so the cheapest seats are likely selling well already. You could try a broad search, say til 1 June, find the cheapest ticket and then try to standby for an earlier departure. Or two, just bite the bullet and book the best fare for the dates you want to go. But “standby fares” went the way of the dinosaurs with paper tickets 😉
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