The Beyerdynamic DT-990-Pro-250 is the highest fidelity headphone you can buy for under $200. I own this one, and it’s amazing. I’ve spent a lot of time examining headphone lists created by audiophiles, and this headphone was recommended many times.
It’s an open headphone — this mean others around will be able to hear your music. I recommend using a mini-amplifier with these headphones, you intend to use them with a portable MP3 player. I have a Fiio E10 USB DAC Headphone Amplifier and it works well.
Other Options
The only other headphones that can compete at this price level are the Sennheiser HD 598 Headphones and the Audio-Technica ATH-M50 Professional Studio Monitor Headphones.