All of Ray's portable amps can drive the HD650 no problem.
He actually uses the HD600 when designing his amps.
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Even the cheapest Ray Samuels portable amps happen to cost as much as OP's budget... That's something to think about.
http://www.raysamuelsaudio.com/purchasing
First, the HD-650 will require a decent amp to drive them, whether it's a portable amp or not. The E6 would be a joke with the 650's. You'd probably have to spend as much on an amp as you would spend on the 650's (around $400 to $500 plus). And what about the need for a DAC?
Do you listen to Lossless files (FLAC), or do you listen to inferior low quality MP3s ?
From what you've written are you just interested in loud ear-damaging volume, and sound quality is not that important. If that's the case and you use MP3's (or even worse YouTube audio), then just get any $20 headphones and use your Fiio E6.