Always an interesting question … and some specific discussion around actual systems at this level is going on here (well worth reading).
Beyond that, to answer more in the spirit that I think you’re asking …
While headphones tend to have the biggest, most readily discernible, differences from each other, even a “perfect” headphone can only, at best, reproduce what it is given; which is the underpinning for the “source first” argument. At the same time, the best source or amplification in the world can’t make a headphone operate outside the laws of physics, and the more audible differences between transducers is the basis for the “headphone” first position.
A low-impedance, high-efficiency, headphone, can be driven straight from portable devices and adding an amp may not get you very much initially. But the better the headphone the sooner it’ll start to show up weaknesses earlier in the chain. And, oddly enough, outside the really exotic stuff, higher impedance, especially for dynamic cans, tends to correlate with better sound (it might not be the cause, but it occurs often enough to comment on).
Another way to look at this is that the law of diminishing returns sets in much more slowly for electro-mechanical devices (i.e. headphones and speakers) than it does for pure electronic devices. This means that you get a bigger bang for your buck in terms of improvement as you spend more on the headphones vs. spending on an amp or DAC (provided you have adequate, clean, power to drive the headphones in the first place).
The real, technical, performance difference between a $100 DAC and a $1,000 DAC might be 10% and you may have to pay close attention to hear it at all. With an amp, it might be 20% and consequently a bit easier to discern. With headphones its likely to be an unmissable difference and might well be “twice” as good.
Some headphones “scale” better than others. The ~$200 HD6XX will keep improving even with amps up to a couple of thousand dollars. And you might find a $400 headphone, that stops scaling before you get beyond even a basic $100 amp like the Schiit Magni or O2.
So … if it were me, I’d put the bulk of the budget into the headphones initially. Exactly how much would depend on whether an amp and/or source purchase was going go be needed. But probably not less than half, and more likely as much at 60-65% - though availability of some, special, models from time to time can skew that (e.g. the HD6XX at $199 are basically the same as the HD650 at it’s typical street price of $319).
I generally wouldn’t spend more than about 20% of the budget on an amp or on the DAC at first, in that case. Which would give you about $300 for the headphones, $100 for the DAC and $100 for the amp. There are many ways to spend that. If you have an existing source, then I’d divert the cash to the headphones first.
Progressing from there is about priorities and balance.
Buying used helps a ton.
As does buying as infrequently as possible; a smaller number of larger upgrades will save you money in the long run. It is very easy to get trapped in the world of side-grades if you’re too eager to upgrade but don’t really have the budget for a meaningful change.
Another pitfall to avoid is to spend money on multiple inexpensive “tweaks” that could get you a bigger upgrade if the same expenditure was made in a more concentrated and considered fashion. No amount of “USB gadgetry” is going to turn a $100 DAC into a $1,000 one. Such things should, ideally, be left until you’ve taken the major components in the chain (DAC, amp, headphones) as far as you think you’ll go. And the only exception I’d make there is if such a gadget was fixing an actual problem (audible interference or insufficient power on a USB port).
Anyway, that’s a bit more long winded than I intended … but hopefully it’ll add to your thinking usefully.