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For doom on a budget, this seems to be a winner:
https://www.andertons.co.uk/tone-city-matcha-cream-fuzz-pedal/
Pretty much the only time I’ll watch an Andertons video is when it’s Oz doing doom stuff. The Matcha Cream is a tremendous doomy muff style pedal and beat out some very expensive / famous fuzzes.
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Silicon Fuzz Faces are great, they are brilliant overdrives (believe it or not) at the low gain end with the volume slightly rolled down and can get very woolly cranked up. There's different shades of it though, from the more traditional ones to the more doomy stuff like the DAM Meathead, which has no gain control but responds well to the guitar's volume. Clones are available too (and very easy to build if you're inclined).
The variation of it known as the Axis is a great middle ground and very dynamic but somehow not the most popular so not easy to find cheaper clones.
Tone Benders are definitely rowdier and somewhat more lo-fi. They do the Tone Bender thing great (duh) but maybe not what you're after.
Otherwise go green Russian Muff for that Kyuss-like fuzz tones, just don't expect it to clean up.
Not quite a fuzz, but I love Rats and my T Rex Mudhoney, they do get fuzzy at the high gain end but do a lot more too - if you can get your hands on one to try it might.
Finally, at the very cheap end of the spectrum, the Behringer SF300 SuperFuzz and the TC Rusty Fuzz are great sounding fuzzes, both inspired by Boss fuzzes which in turn are respectively inspired by old Shin Ei fuzzes (the Behringer/Boss FZ2) and the Fuzz Face (the TC/Boss FZ3). Somehow, these two might be amongst your very best bets come think of it, and yours for no more than £50 for the pair...!
Recommend me a cheap first fuzz pedal? Baffling array of choice it seems, especially amongst the little ones on Amazon/reverb etc.
Going into a small tube amp, if there’s just a richness at one end and doom at the other that’d be great.