here are two that I've been pondering:

1) "it's the absolute opposite of a no-brainer" - given that a no-brainer is defined (by M-W) as something that requires a minimum of thought, what would the exact opposite word be?

2) consider the lyric for I’ll Do Anything by Jason Mraz:
“We can cool in the gang if you’d rather hang
Ain’t no thing. I can be lucubrious with you.”

Urban Dictionary suggests calm, cool, relaxed; the closest Latin root I can find is lūcubrāre , to work by artificial light [cf. lucubrate] - what's going on here?

2a) or is it related more to luculent, clear in thought or expression: lucid

1) That's a real poser.

2) Mraz just thought the word sounded good and had no idea what it meant having only seen it in a context from which it was not trivial to understand its meaning. Or either it wasn't Mraz that came upon this word. Lucubrious has an OLI of 1, and that's from Urban Dictionary.

I can be lucubrious with you.

At first I thought it might be a misspelling and mismeaning of lugubrious, but who knows?

I guess we can say (at least for now)

lucubrious, adj. chill; relaxed; calm in an upbeat manner. Etym. unknown.

My immediate tought was, "He's got the word wrong - he really means 'lubricious.

I could be wrong - not that I'll usually publicly acknowledge such an event .........

this from songmeanings.net

lacubrious is a made up word by hip hop artist bushwalla which means chilled and relaxed

Going back to tsuwm's original query:

[here are two that I've been pondering:

1) "it's the absolute opposite of a no-brainer" - given that a no-brainer is defined (by M-W) as something that requires a minimum of thought, what would the exact opposite word be?]

Would this be a "full-brainer"? - meaning that the question/problem is difficult and therefore needs the full power of whatever brain you have?

edit
For instance, if you are acknowledging that you have limited experience in solving a particular problem, you might say, "!You might find this easy, but for me, it is a full-brainer."

If no-brainer stands for no brains involved the opposite could be all-brainer (all brains involved).

(all hands on deck)

I can live with that one, Bran.

I can live with full-brainer too.

Well--I'm not sure, tsuwm, if you meant an opposite match to the term in which instance full- or all-brainer would work, I guess, though I don't like either of them very well, or if you meant a word for an opposite meaning (doubtful) in which case poser, conundrum, etc. would do.

Let's see--no-brainer means no (or little, as per the official def.) thought is required. So the opposite would need a word for full amount or maximum thought. ... ! Brain-buster?