There are always a few things that end up on my equipment list with just about any character I play:
1) Three large sacks. One of my earliest D&D games had the group find the Big Room O' Treasure, a Scrooge McDuck sized hoard. What we realized is that with all our 10' poles and 50' of ropes, we didn't have anything that would help trying to carry out a bunch of coins. We ended up filling up our backpacks, our boots, and dumping out all our normal gear. By the time we got back to that room, all the treasure had of course been looted or otherwise moved. Since then, I've always written down "Three Large Sacks" on just about every D&D character sheet I've had.
2) 50' of rope with grappling hook. Always good to have rope. I consider 50' to be about the minimum, if I've got the encumbrance to spare I'll usually lug along another 50'. If I have a donkey, it probably has 200' of rope on it.
3) Bedroll and winterblanket. Even though enviromental conditions are rarely mentioned (unless they're part of the module), my characters try to stay warm at night. Sometimes I tack on a sheet of canvas (for rain shelters) or a hammock (more comfortable than sleeping on the ground).
4) Flint and Steel. Depending on the edition, this might be a tinderbox, a flint and tender, a sparker or matches. But again, even though survivial issues rarely seem to be brought up anymore, I tend to have something that can start a fire.
5) Sunrods. The fantasy world version of glowsticks. Nice to have, even if you only use them to find out if the bottomless pit is really bottomless.
6) Crowbar. Because sometimes the rogue blows his Open Locks roll. Or opens his Blow Locks roll. It can happen to anybody.
7) Waterskins. Despite the fact that Create Water is a first level spell and no one has trotted out dehydration rules in some time, I always make sure my character has a gallon of water on him. Plus, if I get thrown off a boat, I can empty out the water, inflate them and use them as floaties.
8) Back Up Weapon. I like to have a decent but nothing special weapon to use when we're inevitably fighting monsters that break/rust/dissolve or take weapons. If I get high enough level or we're swimming in them, this will be a masterwork weapon, otherwise it'll probably just be a 5gp light weapon. This will typically be of a different damage type than my character usually uses, for those situations where a specific damage type is needed. It'll also be something that has a non-combat use, like a hatchet or a hammer.
9) Pajamas. If my character wears heavier armor, I usually keep a set of Leather Armor just for sleeping in. That way, if we're attacked in the middle of the night, I got something. As I go up, this might be a mithral chain shirt, with magical bonuses if we've run across one.
10) 3 yards of Common Cloth. I used to always take this for wound dressings, even though D&D has pretty much the most available and convenient healing magic in the imaginary world. I got the idea from a long ago Dragon Magazine article (circa '89?) and it always stuck with me. You could also use it as a towel, and you should always have a towel with you.