So, how about I get into the first time I realized I'd experienced what I'd now call deep immersion in character...
It was in a live action game, rather than tabletop. It was on Sunday morning of a larp convention, and the con ran a sort of project each year, with a game written during the convention.
The scenario was set in an art museum, during an attempted robbery gone wrong, turned into a hostage situation. I was playing an unarmed, off-duty cop who just happened to be at the museum that night.
My character, like the rest of the hostages, got hands tied behind his back (simulated by my holding my hands behind my back). And I'm sitting there wondering what to do. One of the GMs made rounds among the hostages, and when he came around to me, after a couple of questions I determined that the only item I can reach this way was my police badge, which is in my back pocket. I asked if I could use the metal edge of the badge to wear away at my bonds. The GM thought this was reasonable, but I had to keep wiggling my wrists behind my back, as if I were trying to do this, for 10 minutes. If the thieves noticed, well, they noticed.
That was a long ten minutes. While I was doing this, the pregnant character across from me started going into labor. And the Mayor's wife (also a hostage) who had always wanted but couldn't have kids, went into an emotional crisis over that, so the thieves had to come right next to me while I was still trying to free myself.
And my real-world heart rate became elevated. And I had to pay attention to my real-world breathing pattern, to keep it normal so I would not be noticed....
That real-world physiological impact was a sign of solid immersion into the emotional state of the character.