First, that depends on your ears. If your hearing cuts off at 15KHz, phones that reproduce to 20KHz are a waste of money for you - you'll never hear any difference between them and cheaper ones that only reproduce up to 15 KHz.
Second, you didn't tell us what style you want. Earbuds in the ear? Earbuds on the ear? Earbuds over the ear? Headphones around the neck? Over the head headphones? How large a "moose rack" are you willing to walk around with? (I have some great SoundDesigns that weigh a few pounds and make you look like you're about to go back to the planet you came from - but they reproduce anything any human being can hear (and a lot we can only feel).
Sennheiser makes some great products. They're not cheap (which for Sennheiser is "under $300"). You can get some pretty good earbuds for under $20.
You have to narrow it down a bit. "The best earphone" is unrealistic - it exists in a laboratory and to reproduce one for you would cost more than any phone made. And you probably couldn't hear any difference between it and a good $20 or $30 set of earbuds. Remember, the fidelity of digitized sound is already compromised, since you're only hearing samples of the sound at frequent [very frequent, like 96,000 times per second] intervals, not the actual continuous sound that it was originally. Then you're hearing that reproduced in a player that's probably only reproducing half of it (48,000 times per second). The distortion produced by that process alone makes the sound itself not "the best", so the best earphones, reproducing exactly what the player is playing (which is physically impossible), aren't reproducing "the best", only a fairly close approximation to it.
Most earphones have no problem reproducing highs with very little distortion. (No earphone can reproduce sound without SOME distortion.) The problem is at the low end, the bass range, and that's where the price starts climbing. The lower the frequency, the longer the wavelength, so the larger the vibrating reproducer has to be to produce any volume and decent fidelity. The diaphragms on the largest headsets are only a couple of inches in diameter - the wavelength of 20 Hz audio is over 56 feet. So, at best, earphones that reproduce "good bass" are faking it. (There are many methods of increasing the bass output of earphones - none of them do justice to the sound - which is why people who want the best audio buy such large speakers for their sound studios. A really good 15" woofer (I'm not talking about the "superwoofer" toys that replace fidelity with "loud") reproduces down to almost the lower limit of human audibility - cleanly. (Meaning that the amount of distortion it introduces is so little that only a good musician or audiophile can detect it.) Larger speakers can reproduce even lower frequencies just as cleanly, and low frequencies (16-20Hz) so cleanly that most of the best can't detect the distortion. (Instruments can.)
So what was that you wanted again? A $900 pair of Sennheiser headphones that will tire your neck in 5 minutes - but reproduce great sound? Or $15 Koss earbuds that will reproduce fairly well down to 60 Hz? (Walmart carries them and they sound pretty good - not pretty good for cheap pieces of plastic, but pretty good. Nothing like a Sennheiser headset, tough.)
Please bear another thing in mind. Driving a lot of sound directly into your ears damages the cilia in the inner ear. It's not apparent immediately, the damage is very slight, but a dead cilium doesn't regrow into something that works. Eventually (around the time you reach 50 or 60) you notice a ringing n your ears, usually around 4 Khz. That's the "sound" of no sound. Your cilia above 4KHz are basically "dead", and your hearing ends there (you can't hear anything above 4KHz, and that includes the parts of sound that we don't hear, but that modify the parts that we do), so your brain fakes a sound at the missing point. (You know the letter "s"? You'll never hear that hiss again. And the difference between "p", "b", "t" and "d" becomes just about non-existent. If you can't see the lips, you have to guess at the word.) As you get older it becomes louder, to the point that you can hear people talking, but unless you learn to read lips, you can't understand them. (It can also drive you batty, hearing this "singing" in your ears 24/7/365. You keep thinking you "heard" something, then realize it's only the ringing you always hear.)
BTW, that's one reason a lot of old people become recluses or antisocial - they have no way of communicating with others. Sitting around the dinner table, trying to join in the conversation is like watching TV with the sound turned off - and there's no "closed caption" for real life. So don't yell at grandma for sitting there mumbling while the family has a conversation - make sure she can see your lips when you talk.
I love people who sit there with their hands over their mouths, talking to me. I have a little sign made up, with a font that's almost impossible to read, that says "as soon as you're ready to let me see your lips, we can start having a conversation". When we talk I tell them that reading the sign was easier for them (and it's not easy) than trying to understand one or two words of what they said when they had their mouth covered.
This isn't some "science type" trying to forecast your future - this is a 71 year old man who spent years at consoles with headphones turned up, so he could hear the most minute error in the sound. And now wears a tiny device over each ear - that costs $4,000 - each. And if a drop of water gets into a hearing aid like that because you're caught out in a storm? There's no water damage warranty - you buy a new one. Only to have hearing about good enough to fake being able to actually hear. If not for Bluetooth, cellphones would be useless for me.
Protect your ears. Once the audiologist shows you the graph with the 100db notch at 4Khz, it's too late. And NO headphones can make up for the loss. And at the age of 20, I could HEAR where the sync pulse kicked the horizontal oscillator on a TV set. That's hearing a change of phase of a 15 KHz triangular signal. Not hearing the signal, hearing the point at which each cycle of it changed phase. (Those of you who remember TV sets with vertical and horizontal knobs - I could set the horizontal by ear, I didn't have to be looking at the screen.) Now I couldn't tell you if the set were turned on or off if I stuck my head in the back of the cabinet - not by the sound.
Please, people, protect those ears of yours. The neighbor down the block doesn't have to hear what's in your earbuds when you're in the house with the windows all closed.