Gygax wrote up a sample in the 1st Ed DMG - Pages 96-100. There is also some lead-in and outline discussion that sets the scene for the sample play.
It narrates, multi-voice in the first person, an initial exploration and beginning combat in a low level dungeon. You only get the voice of the DM and the party leader. Apparently Gygax's method involved streamlining things by having a party leader call the shots for the party. Something I've heard of but have never actually seen.
Gygax also gives a quick "how to" for running combat on page 71 of the same book. He also discusses DMing encounters via prior set-up and prepared "monster" actions on pages 104-105 for some hypothetical RPG scenarios.
I suspect there are loads of other examples out there - stuff used at early GenCon events, early Dragon Magazine articles, and the like. But the 1st Ed DMG gives you something straight from Gygax himself at the inception of 1st Ed.
This talk, between the 20-25 minute marks, describes the general atmosphere early in the game's development - both Gygaxian and Arnesonian style campaigns circa 1972. It's David Ewalt presenting "Of Dice and Men: The Story of Dungeons & Dragons". The whole talk is worth watching, though.