I bought my DT770-80 for about $125 last May on Amazon. Great headphones. Some of the best soundstage for closed headphones and they still retain decent bass response which I think seem to work against each other in headphone designs. Beyer has figured this out at a good price. For the price, I feel they are a great deal.
I power them with a Cayin C5 out of my phone (S5) which I think has a decent DAC. I've listened to them out of my Sony Vaio laptop and switched to the S5 because the DAC IMO sounded terrible. I also use the C5 bass boost when I'm in the mood which I think works well with them because of the clean lower mids and upper bass, so with the C5 bass boost on, I think it's complements the signature nicely adding that extra fullness. The C5 bass boost doesn't work well with all headphones. I picked up the C5 for about $100. It's nice because it's very portable, lots of power, bass boost, decent battery life and some people even use it during charging like a desktop unit.
If I were concern about the DAC, if consider the Schitt Fulla 2 at $99. I haven't heard it BUT it's small, compact, a combo unit, decent power specs, and is transportable, possibly portable depending on your definition of portable and fairly inexpensive, relatively. I'd check the thread for synergy with the DT770.
If you ever decide to upgrade later, the Fulla 2 might be a good travel unit due to is capability and form factor. I have the Grace m9XX which is similar but higher end and I really dig its form factor and capabilities for this very reason. It's transportable IMO. I use it primarily as a desk unit and if I couldn't afford it, something like the Fulla 2 looks attractive.
To be clear, I haven't owned or heard it but read up on it plenty enough that I think it's worth investigating.
If you have a bigger budget and want something more portable, the Oppo HA2SE looks like a great option.