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Silicon diodes have a minimum forward drop of about 0.6V so that's a practical minimum for Si devices. Germanium diodes have a minimum drop of about 0.4V so that's about the minimum for Ge devices.
I have a Sony AM-FM pocket radio which operates from one AA battery and will work down do near 1V while driving earphones, which I consider somewhat amazing for an analog IC. That's probably a reasonable minimum for a practical circuit today (but most require more than that).