More than price I read "Budget" as portable.
For the most part I use these diamond paddles.
This is my EDC sharpening tool roll. Unless I'm just out to dinner it is with me in my back pack.
They come in a pouch (shown below) and they sell individual ones to fill in the gaps in grit.
They sharpen anything from my pocket knife or box knife to drill bits and even masonry and carbide bits. Thin so they can get into the flutes. I've gone down to as small as say 2mm drill bits getting clean twin spirals of swarf afterward

Very portable / thin / light / super durable / don't require lube or water to rinse off.
Being a man engineered water stone freak I MUCH prefer water stones and . . . basically . . hate diamond sharpening. I use diamonds when I have to but . . .
So I have taken to cutting the ends off some of my old water stones to make smaller, more portable hand held stones. As with Michael Christy I have come to appreciate holding the stone in my hand.
So
for the medium alloy knives I go for these. Old Nortons mostly 700 to 8,000 ( have a Shapton Pro 220 that is not in any photos yet. (((don't waste your time with the Norton 220)))
For the basic low alloy and plain high carbon steel the natural water stones are pretty handy and nostalgic. The leather slips are labeled with which stone came out of it so that is very much appreciated. I have hardly used these but fun to have and nice and thin so more compact for carrying around. Might want to add a strop to this until you get the hang of deburing and that super fine Translucent.
Oh !
I almost forgot. The Spyderco Ceramic Triangle rods hand held.
I pretty much only use it on my M4 for an ever so slight touchup. It won't debur some of the wimpy stainless. Great for unrolling those rolled edges and getting an edge serviceable until you can take it to something that will cut all that off.
Gets into dings in the edge to !
EXACTLY WHAT YOU WANT FOR RECURVES ! ! !! (more than one grit though)
Call that a "System" or a bunch of crap stuffed in a bag.
For a true System I don't look at the price tag.
When I want Formula One cutting edge performance I look to the Edge Pro Apex, Shapton Glass stones and or Die Maker's Diamond Matrix stones. Formula One ain't cheep but it's fun ! And worth it ! ! !
There is just no comparison. A fairly impressive muscle car is OK . . . but put it beside the latest all out Formula One car and it's all over before we get to the first corner.
And after the first corner ?
Well there's always the fire crew . . . I hope he's OK . . . did you see how many times he flipped and rolled ? ! ? !

Need a little better chassis for the cool guy stuff.
Oh look at that ! The knife stays in place with no clamps !
How can this be ? ? ? ?
(in this photo not even a magnet)
Imposssssiiiiiblllllle and yet there it is.
