I really have no fav live recordings and typically avoid them but the first time I heard electrostatic speakers at a high end shop where I was placing an order on a Mirage 5.1 set. They had 2 huge monolithic looking speakers sitting in the center of the 25'+ x 40" room with a one step up stage at one end like a church, I'm pretty sure it was a church at some time. I'm not really a fan of the Eagles and never owned anything of theirs or any band members recordings but when I heard the Eagles Live (it was new then) it sounded like I was right there and was ready to cancel my order and spend the $8k for them since that was a deal at half price the typical $16k sale price since this was the last set and the demo set.
My GF talked me out of it by using common sense at that moment but it set me on a quest to someday get me a good set of electrostatics, and just like that, only 20 years later, I had a bunch of them, so the Eagles Live has a memory that lead to my understanding that there are good speakers if you can afford them or willing to make your own. The difference between $1k speakers and $15k speakers is so vast that they can't be compared on any level except for the price paid. If it was any other speakers I probably would never have remembered the Eagles live was playing.
Otherwise I would have said Peter Frampton Live, it's unusual that a live album propelled him into a rock star, I can't think of any other band that became popular from a live recording and I never heard studio recordings.
Kiss Alive always delivers no matter which country they may have been while recording. Kiss always made clean recordings with perfect production.
Other live recordings are always messed up in some way and the bands sound like they're having a hard time playing their own songs.