Several parts combine into one character.

Pictograph: the silhouette of a tall watchtower or multi-story pavilion seen from the side. The roof at the top, two stories of structure below. The tallest building ancient scribes regularly observed became the prototype for "tallness / height" itself. Identical across 繁體 / 新字体 / 简体.

Mandarin: gāo, level 1st tone. 高 covers physical height, abstract elevation, and price. 高 (gāo, tall / high), 高兴 (gāoxìng, happy — "high spirits"), 高级 (gāojí, high-class), 高速 (gāosù, high speed), 最高 (zuìgāo, highest), 提高 (tígāo, to raise / improve), 身高 (shēngāo, body height). The collocation 高兴 — Mandarin's most common word for "happy" — comes from "elevated mood".

Japanese: on-reading コウ (kō) for compounds — 高校 (kōkō, high school — abbreviation of 高等学校), 最高 (saikō, supreme / the best — common slang for "awesome!"), 高速 (kōsoku, high-speed), 高級 (kōkyū, high-class), 高山 (kōzan, high mountain), 高血圧 (kōketsuatsu, high blood pressure). Kun-reading たか.い (taka.i) is the everyday adjective — 高い (takai) means BOTH "tall / high" AND "expensive". Context disambiguates: 高い山 (takai yama) = tall mountain; 高い時計 (takai tokei) = expensive watch. The double-meaning is identical to English "tall and pricey both go up" thinking.

In Japanese 高い is the opposite of two different words: 低い (hikui, low) for height, and 安い (yasui, cheap) for price.

高 also forms many surnames: 高橋 (Takahashi, "tall bridge"), 高木 (Takaki, "tall tree"), 高田 (Takada).

Memory aid: a watchtower's silhouette — the original object that defined "tall".