After listening to the show I realized I forgot about whole categories of geocaching gear, mainly because they are a part of my day-to-day life and get used regardless of geocaching or not.
I carry several pens in my pack and two in my pocket. I am partial to the Pilot G2-mini pens as they are cheap (about $1 each) and short, so they fit into packs and pockets, however they still write well (they don't feel cheap). I lose pens frequently, so while I would love to get a Fisher Space Pen or something similar, I suspect I wouldn't have it long enough to fall in love with it.
Led Lenser P7 tactical flashlight (200W). Tough, and durable. Its focusable so the sharp beam comes in handy. I sometimes use it during the day in slightly shady areas to help focus my attention and make metal bits shine. (Try it, it works!)
A multi-tool of some kind. I almost don't think it matters which one you get, so get the one that works for you, but having some sort of tweezer/pliers, knife, and small screwdriver has come in handy many many times. I use the Leatherman Wave, mainly because it was a gift - not cause its necessarily "the best". However I have built out a significant portion of a Google Datacenter just usingt hat tool, and a RJ45 Crimper, so its rugged and versital. I keep it on my belt so I always have it, even when not geocaching. Like I said, get the one that works for you, but if you are not carrying some sort of multi-tool/Swiss Army knife, you are missing out on one of the joys of the modern world.
Tilley Hat. Rugged, dependable, and made with british Hardware and Canadian Persnickitines (its on the label!). I rock the TWC6 Outback model. It floats, is waterproof, light, has extra straps to keep it firmly on your head, and is crushable. I have folded it up and stuck it in my pants pockets, then unfolded it hours later and it still looks good. Its also guaranteed for life.
Cargo pants and shorts are also loved, tho I am not sure it qualifies as "essential geocaching" since its all I wear. Even my bathing suit has cargo pockets.
Bug Spray - 40% deet keeps even ticks away. I have learned to get brands that specifically mention ticks on the label, as I've used some that didn't keep the ticks away.
A good pair of hiking boots is also necessary, tho if I am only going out urban caching I will go without. However odds are if I am planning on a day of caching, even if its all expected to be urban, I wear hiking boots - you never know what you will get into, and boots go more places than shoes.
A pad of paper, and a camera (both for keeping notes on multi's and puzzles) round out my list. I use a Rite-in-the rain pad, and my camera is a Canon S100 point and shoot (tho for the purposes of keeping trail notes my cell phone camera gets a lot of use as well).
I keep all my gear in a Timbuk2 messenger bag - that thing is rugged as... well... very rugged things. I love it.
So thats my list. Do with it as you will.
