Anybody know a good way to get this faucet apart?

Pry the logo off with thin blade, from the edge (try not to chip). It should pop out and expose the set screw. Go from there.

That's a single lever faucet; will have a cartridge of some variety, not washers...

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The first thing I'd recommend would be to figure out who the manufacturer is and see if can find anything about it from them.

Failing that, photographs too out-of-focus and not closeup enough to tell much. Might be something under the top emboss; see how it's on there.

Other than that, the top is a press fit if not as it can't be threaded as the handle prevents rotation.

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Try the cowling first. It should be easy to get off. be gentle

Or just do yourself a favor. Run out to one of the big box stores, get a decent faucet, rip that one off that you have and install the new pretty one. Bubba

Looks like an early Moen faucet. Try lifting the entire cap straight up. Once that comes off, the lever is held in place by a plastic leverage bridge that pulls the cartridge up and down. The lever and the bridge are usually held in place by a screw.

If I remember correctly, you will find a round knurled knob that unscrews. Once that's off, you can remove the retaining clip and pull out the old cartridge. Sometimes they're a bearcat to remove. In that case, run out to Home Depot and buy a special puller tool for Moen cartridges. They run about $15. That turns the job into a piece of cake.

Buy a new cartridge and lube it with plumbers grease. Pop it in and re-assemble

I doubt its a moen but thought I would mention that a bunch of moen faucets built 6 or 7 years ago were defective and started leaking under the sink. Moen will send you a kit to rebuild it (not just a new cartridge) or sell you the new model at a deep discount. The defect involved a anti siphoning mechansim that was required for a short time before the gov gave up on it,