Here's the thing, there are a few cheaper headphones that make things similar to the HD800 but not all at the same time, that's your trade off here. You'll have to pick those things you're willing to trade.
DT880 is closest to the Sennheiser HD800 in terms of sound signature, very close indeed.
You'll get most of the air, treble extension and quality, slightly recessed upper midrange, neutralish extended bass, and so on.
This is the poor mans HD800 in my book.
Soundstage and clarity in the midrange and bass are the main trade offs. It's not that the DT880 is bad, but it's not very close to the HD800 in those particular regards. If you are a soundstage-head then you might look elsewhere.
AKG K702 will provide a similar experience in terms of soundstage size, HD800 projects bigger objects and thus it's a bit more intimate, but both K702 and HD800 are top picks for soundstage and imaging despite the slight difference in presentation. Other strong point for the K702 is the bass and midrange clarity, it surpasses the DT880 in this regard and it's not far from HD800. The AKG is very precise across the board, but it's not as extended as DT880 or HD800 in the treble region and thus it lacks some air when compared side by side. That and the slight bump in the upper midrange make the AKG sound markedly different than the HD800 in terms of overall balance.
HD600s render vocals very well and it reminds the HD800 in that regard despite having a very different sound signature (mid-bassy, forward upper mids and soft treble). You can pick each an every word in a more clear way than both DT880 and K702. Other than that HD600 is a totally different headphone, softer, not analytic, very small soundstage and so on. Wait, HD600 smells similar to the HD800.
Hope this helps!
