I'm running a 4e, duet campaign in a non-magical world. Because the hero is travelling alone, he sort of needs "extras" so he doesn't get killed all the time. It's a real struggle and I don't claim to have it figured out, but here are some ideas I've been using.
Three items of my own creation:
The Corner Bow - This "L"-shaped crossbow was initially designed to be attached to the side of a saddle. The rider would reach down and pull the trigger, sending a bolt straight ahead, into the line of enemies. When the horse was standing still, the bow worked great. When charging into battle, however, the bow frequently shot the horse in the neck. In any case, you now have a crossbow that can shoot around corners. Perhaps the character finds a small piece of mirror he can affix to the bow. The ranged attack has significant penalties, but it's still an interesting option just before an encounter.
Camouflage - This loose tunic is designed to look like __ (insert common environment like woods or rocky terrain). It gives a +2 to stealth while moving in that environment and +4 to stealth when standing still (to hide).
"Dog" Whistle - This flute has been surprisingly fun. When the character blows into it, the sound is inaudible to most creatures. When you encounter a creature it effects, though, it has dramatic, if unpredictable, results. Roll a random die to determine if the effected creature flees, attacks, drops weapon and covers ears, falls prone and covers ears, falls prone and convulses, or just starts screaming wildly. I try to come up with a new possible effect almost every time the characters tries the whistle. Having a creature, who seemed particularly imposing moments ago, start vomiting uncontrollably makes for a dramatically different and interesting encounter or skill challenge.
Here are a few other, general ideas:
Books - Books don't help someone become better in battle, but if they're carrying them around (hopefully they have mount!), they have reference materials for certain situations. I know, lame, right? A character might find a book on the flora and fauna of the region, history of the kingdom, healing techniques, etc., with the appropriate bonuses to their skills. It also makes for some difficult decisions when the player approaches a certain level of encumbrance.
Traps - dandwiki has a page for Quick Traps (and I think someone mentioned Tagglefoot bags above), which characters carry with them. The character I'm working with has gotten really creative using these, which is fun for them and keeps me on my toes. He can also reduce the number of enemies he'll have to encounter in battle by enticing his enemies, one at a time, down a hallway and around a corner, where... trapped!
Non-magical "potions" - There are plenty of items on Earth - which is non-magical as far as I know - that have effects on people, so there's no reason why there can't be some crushed plant leaves or animal semen or whatever in the world you've created that help or hinder creatures. Peyote, for example, is not magical, but it sure would mess with our hero's enemy if he slipped its powdered form into their drink.
Glands - Anytime any PC I'm controlling kills a creature that might have glands, I cut that creature up trying to find them. This could be your average spider/snake venom (which the PC can use to tip their weapon for ongoing damage) or the glands of a flash or fire beetle. Someone already mentioned using body parts of fallen monsters and that's a great idea. I once ran a PC who killed a poison-spewing alligator monster and, when my DM refused to allow me to get its glands, my character skinned the thing. In the next town, he paid the local tanner to create armor out of it. My DM relented and gave me +2 armor. Obviously, you'll have to introduce these creatures in order for the hero to kill them and remove their body parts.
Alchemy - I don't know if your player considers alchemy to be too similar to magic but, if not, that's an option. This ties in with the potions and glands mentioned above. Here's a decent list:
You could also give vials of smoke or a bottle of some slick liquid that the character can use as a splash weapon - Spy Hunter style. It would give them a controller element in battle, creating areas of difficult terrain or obscured vision.
Cheating??? - I don't know if you would consider this to be cheating, and I apologize if this seems against the spirit of your question, but it just dawned on me that the character in question could possess a magical item which they are not aware is magical. As far as they know, that new set of __ (insert item for which they have an available "slot") gives them a +1 to attack/AC/whatever. What they don't know is that the item is magical and has __ property. This would be a good object to insert into the story if your players fail a detect magic check with what seems like a pretty decent roll.
Then, during combat, when the player wearing/using the object says, "I rolled 18 vs. AC," you secretly add the +2 magical item bonus before determining success. This is more work for you, but it adds the possibility that the character eventually discovers his mistake and is disgusted/kills self/swears off fighting/is angry with companions for not telling him/etc. It could be especially interesting when one of the other players notices that the guy with the (secretly) magical bracers is hitting with 18 when he/she is not. As a DM, all you can do is shrug your shoulders. They'll figure it out eventually. Come to think of it, this could be a fun idea even for a character who doesn't fear magic. And the item wouldn't have to have a bonus property. It could have a penalty. Now I'm typing as I brainstorm, which is a bad idea.
Good luck. Thanks to the other posters here. You've given me some cool new ideas for my own game.