I agree with Steve and others. On my E230 wagon, I had your exact problem and a new Mercedes battery fixed it.

I was surprised how many amps the system must pull to turn over.

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2004 E320 4matic
2000 E320 wagon
-- in the past---
1992 190E 2.3 5-speed! (European Delivery Baby)
2000 ML320
1978 240D - 340k mi