Word for not caring at all about anything

Solution 1:

Since you're asking in a psychological context, are you looking for one of:

Detachment (detached)

Dissociation (dissociated)

Depersonalization (depersonalized)

Emotional numbness (numb)

Based on the updated question, perhaps you're looking for:

Nihilism (nihilistic): Nihilism is often discussed in terms of extreme skepticism and relativism; for most of the 20th century it has been associated with the belief that life is meaningless. Existential nihilism begins with the notion that the world is without meaning or purpose. Given this circumstance, existence itself–all action, suffering, and feeling–is ultimately senseless and empty.

From the Encyclopedia of Philosophy article on Nihilism, though there are plenty of other choices (e.g. Wikipedia).

Solution 2:

The most proper word would be:

nihilism

n [U] [Date: 1800-1900; Language: German; Origin: nihilismus, from Latin nihil 'nothing']//
1 the belief that nothing has any meaning or value:
2 the idea that all social and political institutions should be destroyed

Solution 3:

The condition you are describing is ennui, "a feeling of utter weariness and discontent resulting from satiety or lack of interest; boredom" (dictionary.com). Ennui is boredom on steroids, dissatisfaction taken to an existential level. Oddly, the Oxford English Dictionary reports that ennui can be used as a transitive verb as well—"to affect with ennui; to bore, weary"—although all of its examples are from the 19th century. The adjectival form ennuying is also listed, but described as "rare."

Solution 4:

This sounds like the condition known as anhedonia. The term is more common in medical / psych settings but you do hear it elsewhere.

Solution 5:

I'd never seen this splendid word until today, but I feel compelled to bring it to the public's attention:

From Merriam-Webster:

pococurante - adjective \ˈpō-kō-kyu̇-ˈran-tē meaning indifferent, nonchalant. From the Italian for "caring little"

It may have found its origin, capitalised, as a character in Voltaire's Candide.

There's a lovely non-capitalised usage in an essay on Lord Melbourne, by Abraham Hayward, published in 1858.

… during many years he apparently led a careless, indolent, pococurante life, divided between the gay circles of London and the House of Commons.