Linux users should enable the compose key, it's very useful.

Test it with:

  setxkbmap -option compose:menu

Then press the menu/compose key (next to right control), then C, then =. You get €.

Try compose, 1, 2 for ½.

Compose, ^, 3 for ³.

Compose, A, : for Ä.

It's pretty intuitive for the most useful characters, and easily the fastest way I have of typing the ö, ñ and å in various colleagues' names.

Actually, A: is not valid with the default compose bindings (it is however valid with vim digraphs). In general umlauts or trémas are inserted with the double quote (A").

Worth noting that international layouts (with dead keys) support many of those without the use of compose key (for obvious reasons not your first example – but the € symbol has its own key combination at least where it's commonly used). I admit it might be hard to adjust to caret being a dead key, though.

Dead keys support accents, and a few symbols printed on the keyboard are discoverable, but I find the Compose key is much more intuitive for occasional use.

I don't yet speak the language of my adopted country, so it's better for me to keep []{} etc where I like them in the British layout, and use three keypresses for typing the ø in a (place) name like København.

If I do end up typing lots of Danish, I'll probably map AltGr+A,E,O to Å, Æ, Ø. É is rare, so I'll still use the Compose key for that and German / Swedish names.

I use the default German keyboard layout. Even with dead keys I wouldn't have Nordic characters or the Turkish dotless i and dotted I. I love the Compose key. It's actually making me consider using the US layout because the lack of German glyphs was what was keeping me back.

I use the code behind this web app: http://latex2unicode.herokuapp.com/

It converts latex to unicode where possible. It's pretty impressive how much of Latex can be replaced with Unicode today.

This program translates LaTeX markup to human-readable Unicode when possible.

Here's the default text from that webapp:

Basic math notations: ∵ A͡B + B͡C ≠ A͡C ∴ ∬∜x̅ ξᶿ⁺¹ - ⅜ ≤ Σ ζᵢ ∴ ∃x∀y x ∈ Â

Easily type in hundreds of other symbols and special characters: , ℵ, Œ, ⇊, etc.

Font styles support: 𝔹𝕝𝕒𝕔𝕜 𝔹𝕠𝕒𝕣𝕕 𝔹𝕠𝕝𝕕, 𝔉𝔯𝔞𝔨𝔱𝔲𝔯, 𝐁𝐨𝐥𝐝 𝐅𝐚𝐜𝐞, 𝓒𝓪𝓵𝓵𝓲𝓰𝓻𝓪𝓹𝓱𝓲𝓬, 𝐼𝑡𝑎𝑙𝑖𝑐, 𝙼𝚘𝚗𝚘𝚜𝚙𝚊𝚌𝚎.

Now type in this box and try it yourself. ⌣̈

Careful — some of the Unicode pseudoalphabets won't render on mobile.

No problem on iOS; I see the same thing that I see on my Windows desktop.