You've just said what is needed.
First lot for flight in IMC.
Second lot if needed for navigation. I understand the A380 has a dispensation not to have and ADF or DME, can't remember which but I think the ADF.
Some aircraft, like Cirrus, don't have an ADF or DME. But they can be flown IFR. The legality of some approaches is debatable so you will find you get conflicting advice. You can perfectly well fly without the ADF and DME using a second GPS as the best ADF ever. GPS gives distance to the threshold anyway. But it may or may not be legal, probably not. The solution is to arrange a different missed procedure with ATC.