Comparative form of drowsy; more sleepy, more inclined to doze, or more characterized by drowsiness than something else.
From drowsy plus -er (comparative adjective marker). Drowsy itself comes from drowse plus -y (adjective-forming suffix).
The comparative drowsier shows why English comparative forms matter—you can't just say 'more drowsy' sounds as good as 'drowsier,' which is why we kept the older one-word form!
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