We have a LHBS that will let you brew in their kitchen with expert guidance for $55.00 a batch.
You do need some specialty stuff, but you can sure get away with a lot less than what's in a kit. Depending on what you want to do and how much. I have made a few batches of wine in Milk Jugs, and a LOT of test batches in 1 gallon wine bottles.
(See, Cheeto, you'll fit right in behind the wall.)
As to the original question, you're basically using, at minimum, pots and buckets. My "brew pot" was a five gallon pot I already had. You could use a food-grade bucket for a primary and not worry about a secondary. But the bottling bucket would be the toughest to do without because you have to transfer the beer to the bottles somehow. I guess a sanitized funnel and sanitized large measuring cup (which, I would think, would work faster than a ladle) would be doable but a pain. Reuse empty soda bottles and you don't need a bottle capper, though you'll have to keep them out of the light (doable).
A minimum kit would be bottling bucket, a primary fermenter (plastic bucket) with lid, and an airlock. At a guess, under twenty-five bucks. Sanitize with bleach,
save soda bottles, it could work.
[edit: I assume you're talking about a hardware kit; you'll still need the ingredients to make the beer.]
[edit2: oh, and welcome to the forums here at HBT]
If I were to start over again, I'd probably still go with a basic kit, as long as it was done right. A brew bucket, airlock, cheap capper, some hose. I 'd find a plastic carboy, and a good deal on a brew pot (~5 Gallon aluminum, procelain, or Stainless Steel).
This will get you to doing 5 gallon extract recipes. But, if you look around for a brew bucket, then you can save a good portion of the cost. Get a food grade bucket with a lid. Buy a spigot and drill and mount it yourself.