Mine! (When it's finished) ;-)

All kidding aside, I think every project deserves its own set of fonts. There are of course the 'classics':

Stempel Garamond
PMN Caecilia
FF Meta

All personal favorites of course. I can make this list much much longer, but these three are on top!

“What are the fonts … ?”

I don’t think you’ll ever find a good answer to that. Most of the people on Typophile (if you’re not counting the general ID requesting crowd) are type designers, and we continue to draw new typefaces because we believe they can add something. The “best” font is certainly not a static thing, if it’s even a thing in the first place.

What I'd like to have in my toolbox...

http://www.klingspor-museum.de/KlingsporKuenstler/Schriftdesigner/Poppl/...

"perhaps fonts that will work on almost every occasion/project"

That's a bit of a tall order.

That said, Vignelli is (in)famous for his slim typeface toolbox: Bodoni, Helvetica, Times Roman, Century, Futura

Some versatile and high quality fonts that may fit your question:

Superfamilies: Guardian, Milo, Meta, Scala, Unit.

Sans: Myriad, FF DIN, Optima, Proxima Nova, Whitney.

Serif: Minion, Caslon, Tisa, Novel, Chronicle.

Other: Ex Ponto, Poetica, Verve, Sava, Goudy Text.

Here are some of my favorites:
Transitional–Bulmer
Old Style: American Garamond
Modern type design: Brioso
Sans Serif: ITC Franklin
Script: Ovidus, Trafton

Univers

Aaj zekond Oeneevuers! :-)

A bigger list:

Univers, Garamond, Caslon, Baskerville, Frutiger, Myriad, Minion

I'm not saying they are the best choice for every project though, but they are 'neutral' enough too work for whatever project you throw at it

1) Brushscript.
At least if you want to open a stall in the market, a cafeteria near a railway station or design a logo for a knitwear brand. I once started to make a photos of usage of brushscript in business environment. The trend ir transcontinental, I swear.

2) Mistral.
Preferred by sportswear and vegetable stands in the market. Or a youth oriented magazine.

Forget about 'Helvetica'. Look out 'Univers'. Contax Pro is the new geometric sans typeface series for the 21st century.

http://www.fontspring.com/fonts/type-innovations/contax-pro