There are definitely good “budget” tube amps available, and a few of the less well-known brands started to get discussed here. Beyond that, depending on what your budget is, and what headphones you wish to drive, they can be excellent.
As @Fishcommander and @DavidA have said, the Schiit Vali 2 is a very good starting point and isn’t fussy about headphone impedance. Going up in budget from there, my favorite until you get to $500 or so is the $249 Massdrop Cavalli Tube Hybrid. I’m a bit less of a fan of the Garage 1217 amps, but they certainly have their following.
Now, those all have a couple of things in common which are appealing if you’re new to tubes. First, they’re hybrids, so they are quite a bit more flexible than some pure-tube designs. Second they all just use a single tube, so experimenting with “tube rolling” is pretty inexpensive - as long as you’re sensible about it! But, as hybrids, their overall sound is less affected by rolling since the tube makes up less of the overall signature than would be the case with a pure-tube amp.
The Bottlehead Crack is a lovely little pure-tube amp and it pairs particularly well with the Sennhesier HD600, HD650 and HD800. However, in stock form the bass is a little on the flabby side with the HD650. As an OTL (Output Transformer Less) design it also really needs to be paired with high-impedance headphones … 300 ohms or more (it’s output impedance is 120 ohms, so damping factor is a potential issue).
From there we get to the Schiit Valhalla 2, another pure-tube OTL design, but one with more flexibility to pair with lower impedance headphones. Personally I think this is another that is at it’s best with the HD6XX. It’s a bit more daunting to tube-roll with this amplifier as it runs two pairs of tubes - but the effects of doing so are quite a bit more pronounced than with hybrid designs.
The Lyr 2, which is no longer made, has fantastic drive and slam, and a ton of power, but isn’t the last word in refinement or resolution. The Lyr 3, which is a rather different beast, is a very nice little hybrid amp though. Plenty of power, the ability to put a DAC or a Phono stage in it and use it as an all-in-one, ability to drive pretty much any headphone you like, and a single tube that makes rolling cheaper and easier.
Something of an unobtanium amp is the “Laconic Night Blues Mini” with @Fishcommander references (as the Massdrop EC ZDT Jr.). I’ve not had a chance to listen to this one, but once the Massdrop units are shipping, and the inevitable rash of sales from users that have since moved on, starts (or the unit gets re-dropped), I intend on picking one up to see how it is.
We’re up to $500 now. Let’s push that a little though …
The Woo WA3 is a really nice, pure-tube, OTL amp. Again best used with high-impedance headphones, it has a bit better control over the bass with something like the HD650 than the stock Crack does, and doesn’t have to be self-assembled. At $599 it’s an interesting option. There’s a certain clarity to the presentation with OTL designs and the right headphones that is quite compelling.
But another $100 sees you to what I think is the single best-value headphone amplifier that Woo Audio make, and that’s the WA6. I ran one of these in my bedside rig for a long time. Typically with HD650 or Massdrop Fostex TH-X00 (Purpleheart) and it was unfailingly engaging. It responds very well to tube rolling and is worth spending a good fraction of the amp’s overall cost on doing so. Drop a Sophia Electric Punched Plate 274B in as the rectifier, and signal tubes to taste, and it not only delivers the goods sonically, it looks gorgeous doing it.
I’ve heard a few other relatively inexpensive tube amps, including models from Little Dot that were not bad at all, but all were tweaked quite heavily at that time.