Have issue with 13 E350 4matic. Is it normal for A/C compressor to run even though a/c button is not on?

Yes, this is by design. It can be turned off and spin a few minutes.

The AAC module manages the clutch activity with a timer. You can set these parameters under VARIANT Coding.

Typically clutch will go off, less than 5mn after A/C getting commanded off.
Pumping is controlled by variable displacement solenoid such that compressor can be spinning and not pumping.

Would you mind adding more information? How do you know the compressor is running?
1 - Measured pressures using pressure gauges?
2 - Used the scanner to see the electronic control valve and clutch current status?
3 - Visual of the compressor pulley? If so, would you mind posting photo of the compressor with engine running, and with engine off?

So everything in car is off, as soon as I start car, I can gear a high pitch whining noise which is a/c compressor. It never turns off, until I turn off ignition. Reason I think its a/c compressor, sometimes I don't hear whining noise, so I immediately turn a/c button, it flashes and turns off.

Do you have an MB-aware scanner? It would be best if we know about internal error codes, or current live data stream

"A/C button flashing + high pitch sound" is starting to sound like "SUPERSONIC FAN" and not compressor related.

Once you have a scanner, live data + active faults will help your troubleshooting.

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it's the big engine fan running at max speed... this is an indication to suspect A/C sensor.

It sounds like a geared turbine jet engine ready to take off

Yes, thats it sounds like jet turbine engine spinning harder and harder. Is there a temporary fix or can I disable it with fuse until i see a mechanic ??

You don't want to disable all fan cooling on these hot engines.

Try unplugging your A/C sensor harness connector near bottom of condenser coil, that may or may not help one bit.

Did the super fast fan start with very hot weather? If it did you probably have overcharge in the a/c system.

I had this happen. The a/c stopped cooling and fan started running like a jet engine as is mentioned earlier. First thought the charge is low but adding more refrigerant made it worse. Then let some out and everything went back to normal.

The sensor at the bottom of the condenser makes fan go crazy fast when it sees high pressure from overcharged a/c system.

It seems Germans do not have hot days like I (we) do and their charge was too much for my weather.

Very easy to test if this is the problem.

I know- I know- Some specialists will shoot this theory down in a heartbeat...but just telling what happened to me.

My car is a 2010 E550.

If indeed the cooling fan is the source of the whining noise, it will power up the moment engine is running, AC compressor ON or OFF fan will still go top speed.
When engine is cooled overnite, unplug the big power connector to the fan ( easy ) and start the engine and see if the whining noise still there and can it be also turned off along with AC compressor ?
Do note : AC compressor depending on your coding..... sometimes is not OFF immediately with A/C button in OFF, but has time delay.

I have appointment next week to see mechanic, will it hurt car if I leave it and let cooling fan run on high until then.