Answer: Yes!! (Not always though..)

Let's consider your examples again,

Acceptable- "He is the CEO of the (biggest) business (in the history)."

Incorrect- "He is the CEO of (the/our) business."

Acceptable- "It's a TV (making) business."

Incorrect- "It's a TV business."

Informally acceptable- "A business dealing with drugs." (Should be "in" instead or "with".)

So, as Barrie already pointed out, the usage usually depends on what came before and what follows the sentence. The word 'business' doesn't necessarily always mean a company.

When 'business' means a 'company', how do I use the word?
The PAS pump on my Discovery failed fairly spectacularly last week, shedding it's pulley at the side of the road. In order to get at it to replace it, I need to get the fan off, which requires undoing a left-hand-thread nut in the viscous coupling. 

The problem, however, is that I can't stop the pulley rotating in order to undo it (it's on it's own pulley rather than being on the end of the crankshaft). Normally this would be held in place by the serpentine belt, but without the PAS pulley I can't refit the belt...

Any suggestions? I can't get anything in there to wedge the pulley...

Edit: It's the big bolt in the middle of this part (thanks Paulster2 for the link):


And it is located here on the car:


The big pulley spins freely. I've tried using the small bolts that hold the pulley on for extra purchase - two of the four came undone (great, I thought!), but the other two won't - because of the hexagonal shape of the big nut, I can't get a ring spanner on them, and an open-ended one just started to round them...
