As mentioned in the comments, I believe that it's pretty difficult to get a Game Over screen during combat, unless you are just out of Phoenix Downs. There were only one or two times during my playthrough of FFXV where I ever thought I was going to get a Game Over (thanks to Galvanades here, as well), but I didn't. Noctis was allowed to use a Phoenix Down on himself with no HP and a depleted maximum HP bar. I believe that unless you fail to use a Phoenix Down within a specific timeframe after losing all maximum health, you will not get a Game Over screen.
According to the wiki page on status conditions, you only have a few seconds to do this:
If a party member's max HP falls to 0 they are Down. If Noctis gets Downed it spells a Game Over unless he is revived within a few seconds. Downed party members can be saved with Phoenix Downs.
Additionally, if you are playing on Easy mode, it's even harder to get a Game Over screen during combat, as Carbuncle will show up once per combat to prevent what would otherwise be a Game Over.
In Easy Mode Carbuncle appears when the player enters critical condition to heal Noctis, and can appear in both dungeons and the open world. He can appear once per battle. He can also randomly appear in the background of photos taken by Prompto.
However, there are a few other ways to get Game Over screens.
From experience, the most common way (after beating the game) to get a Game Over is to fail to land the Regalia Type-F properly, as failing to land gracefully almost always ends in a Game Over screen.
Another way, mentioned here is to fail to follow Ardyn closely enough when he is leading you in his car.
There’s a part in Final Fantasy XV where you have to follow Ardyn in your car, the Regalia. It’s meant to be a simple part of the game, and if you somehow fall behind, Ardin will slow down, so there’s no way you can lose him. However, some players have experienced a glitch in which Ardin will just shoot off at full speed and they lose him and get a Game Over. Unfortunately, there’s no known solution to this bug yet. Hopefully, it will be fixed in the future.
I didn't personally encounter this, but it sounds pretty funny, aside from the whole making you hit a Game Over, and all.