This is an area of _____ dense population.

The blank requires an adverb that modifies the adjective 'dense' to describe population. 'Extremely' means very or to a high degree, and it appropriately intensifies 'dense'. 'Totally' suggests completeness which doesn't fit naturally with 'dense'. 'Tangentially' means in a tangential or peripheral way, which is inappropriate. 'Languidly' means in a slow, relaxed manner, which doesn't fit. Hence, 'extremely' is the most appropriate choice.