A compose key is very useful if you’re a typography snob — as many of us who studied mathematics and ended up learning TeX probably are… I haven’t been paying attention to exactly what I’ve typed with it lately, but I habitually use symbols like these on autopilot and they seem to render OK on any device that someone reading my writing is likely to be using:

≤ ≥ ≠ × — – “ ” ’ ° … ¹ ² ³ ™ • ♣ ♢ ♡ ♠

If you work in languages other than English but have a standard English keyboard layout, a compose key is handy for typing accents and non-English letters/ligatures too.