{To Follow}

Capital Eszett.
An old character Unicoded.
Much discussed at Typophile.
http://typophile.com/node/33647
http://typophile.com/node/48746
http://typophile.com/node/82806
http://typophile.com/node/85692
http://typophile.com/node/84295

Please, define “recent” ;-)
Post-Euro?

For Cyrillic, you might consider the lowercase palochka.

Only a very rough record:

LATIN
Considerable additions were made, especially for scientific usage. E.g:
• five new, 1E9B to 1E9F (among: ẞ)
• Latin Ext. C
• Latin Ext. D
• u10190 block, Roman numerals
• Monetaria: 4 new, 20B6 to 20B9
• 1DC0 block

CYRILLIC
• several new ch.s in the 0500 block
• new Ext. block A640
• new comb. diacritics block 2DE0

GREEK
nothing new (as far as I’m aware of)

Andreas, I think Tom is specifically asking for what might be called 'emerging characters', i.e. not just characters that have been added to Unicode but which are either recently invented or only now coming into more widespread use.

GREEK
nothing new (as far as I’m aware of)

Until they come out with the new Drachma ;-)

The Sarcmark!

... important enough to show up in standard Latin fonts ...

Oh wait ...

I don't think that there have been major additions in Latin/CE fonts lately. The last really major one, influencing every font, was indeed the Euro sign. Anything else has been quite font specific -- i.e., up to the designer's whims.

If your own font set goes back more than a decade or so, you could put a modern one and an old one side by side and compare them. A modern version of an old font might typically contain more ligatures, small caps, and different digit styles; perhaps some (or lots) of extra accented characters (because Opentype allows more code points to be defined in the font than, say, Type 1 fonts, it makes sense to include lots of them rather than relying on non-spacing accents and having the user combine them themselves).
However, I can predict with some confidence the "basic" characters section will be unchanged.