rein: (noun) a long, narrow strap, typically used in the plural to guide a horse, and metaphorically used to guide people or situations; (verb) to guide; to keep under control (used with adverbs like in and back)
Part of the expression free rein.
reign: (noun) the period in which someone holds power; (verb) to rule as king, queen, or other authority figure
NOT part of the expression free rein.
rain: (noun) condensed atmospheric moisture; (verb) to precipitate said moisture
NOT part of the expression free rein.