Lidl is opening stores in US. Not sure if they are going to do deliveries though.
Lidl does not deliver groceries to your home. If you want a wide assortment, another option would be Bringmeister:
https://berlin.bringmeister.de
(Shameless plug - I work at EDEKA, which recently bought Bringmeister)
AFAIK in Berlin there are multiple choices available. It seems that almost every discount brand and Kaufland have delivery service. As my German is pretty weak I can't find a comparison between them. I think that probably Lidl and Kaufland should have the widest assortment.
Could anyone tell me their opinion about all these?
EDIT: I found a comparison: http://www.bild.de/ratgeber/leben-und-wissen-verbraucherport...