Kontakt instrument favs?

I could write a book!

For strings,

• if you are looking for minute control over details I'd suggest 8Dio Anthology or better yet, their predecessors Adagio and Agitato. If you are willing to spend the time these can sound really pretty.
• If you want something that just sounds gorgeous, and you realize you won't use 100 articulations anyway look at Cinematic Strings and Cinematic Studio strings. I use these more than any of my other string libraries because they have that grand orchestral sound.
• Both collections can cover a very wide array of applications.

Brass is a lot trickier because techniques used for concert, film, or popular styles differ radically.

• For concert or film/theatre projects I've settled on Cinebrass from Cinesamples. I like the sound, and more important, for me, I really like the workflow. They sound great and they are really easy to control. I have only the core version.
• For pop/rock/funk and even big band stylesI use Chris Hein Horns. There is a lot more work involved, but the results are (usually) worth the effort.
• For a pit orchestra I'd use Fable Sounds Broadway Big Band. It is uncanny how well they've captured that whole thing. It is also stupid expensive, and I probably won't buy it until I have a project that specifically needs that sound. Nothing else comes close.
• I really like the idea of Session Horns for trumpet, trombone, and sax, but for some reason it just doesn't work the way I need it to.

Winds are right up there in complexity with respect to application, except most winds aren't used as much in pop settings, except for saxes, which are included in the Chris Hein collection.

• For concert/film work once again I use Cinewinds most of the time because they sound great, and they are really easy to use. Or rather their workflow fits mine?
• I also really like the VSL winds. They are, again, a bit more work, but they sound glorious, and they sit so nicely in a mix.
• As I mentioned, for popular styles I find the Chris Hein stuff works really well. I started with his Compact collection, and even after 'graduating' to the full version I keep Compact around because sometimes that's all I need.

Then there is percussion. You could spend months sifting through all the percussion libraries. Impact Soundworks and SoundIron both have percussion collections that work really well in film settings, maybe in concert settings. But there is just so much out there!

• For rock and roll and pretty much any popular style I use Toontrack Superior Drummer. I'm sure I could get nearly as good a result with any number of Kontakt libraries, but I know how Superior works, so I use it.
• I should mention Jim Roseberry's drum library, except it appears he no longer sells it. I still use it sometimes because I really like the sound.
• If that's not enough you could spend a fortune on cool little hand percussion/found sounds libraries. SoundIron and 8Dio have tons of them, and every one is great. HandheldSound is another really cool company for this sort of thing, and absolutely pick up Perc+ from Modwheel for a wide variety of cool sounds. Impact Soundworks also has quite a few cool little libraries.

(disclaimer: I work for Realitone)
Vocals are among the most difficult sounds to sample. But you may still need them.

• Realitone has two really cool pop oriented libraries - "The Ladies" and "Blue". I won't say more, except that I was a customer before I joined.
• SoundIron, 8Dio, and Cinesamples all have really cool large choir libraries. They have lite versions too, which can be a really good starting point.

Guitars - well I'm a guitar player, so I record all my own tracks, but I also play around with a lot of guitar libraries.
Of the bunch Orange Tree Samples and Indiginus stand out as the best. Two different approaches, but uniformly good results. Orange Tree requires more effort, but then you get more detail. Impact Soundworks has a very cool metal style guitar library that I do use, because I'm not much of a metal style player, and I want to preserve my ears.
 
What else... pianos? There are too many piano libraries, electric and acoustic, none are perfect, and I own several to get around that. I would recommend any of the pianos from Sampletekk, they are among my favorites.
 
What else - I have no idea! If you have specific questions, or need more details let me know.

Don't forget Hammersmith Pro for Pianos...
Keepforest has some nice libraries also for Epic Cinematic things...

Kontakt is definitely "The" sample player for sheer number of 
exquisite libraries available for it.