Coming from this answer and comments under, I realized that all Germanic languages have only the present tense and the past tense. Many also have a full set of subjunctive moods.
To reduce ambiguity, "tense" or "mood" here implies inflected or conjugated (e.g. French j'aurai), not constructed (e.g. j'ai eu). Since other Germanic languages all have subjunctive moods, I think English should also have one, but it apparently doesn't nowadays. Subjunctive mood is formed in various ways like if I were, if I had been and that I should be.
Did an inflected subjunctive mood ever exist in English?