I could not disagree more strongly.

For someone on jessieh's itinerary this would be an awful waste of money. It really cannot be justified.

A.) Cover your train ticket from Narita to Tokyo (Narita Express)
This would be 3,000yen. There are alternative forms of travelling into Tokyo that cost little more than 1,000yen

B.) Cover any travel in Tokyo that you take on a JR line (Which is pretty much any non-subway line), either train or bus.
The value of travel would be no more than 730yen a day (the cost of a JR Tokunai Pass which allows unlimited JR travel for one day within Tokyo)
Which buses??? There are some buses covered by the JR pass but these run to and from Tokyo, they do not include those one can board for local trips within Tokyo itself. Total: 3 x 730yen = 2,190yen

C.) Cover your shinkansen/train travel between Tokyo/Osaka/Kyoto
On a Hikari train that would cost 13,550 yen
You are excluded from travelling on the faster Nozomi train.

And to add some more:
D.) Cover the cost of getting between Kyoto and Osaka
540yen each way

E.) Cover the the cost of getting to Nara
1,560yen for a return trip from Osaka

So - in total:
JR pass - 28,300yen
According to NYCBunny would 'save' you the difference between 28,300yen and 20,840yen = -7,460yen EACH.

(I make the cost of buying a JR Pass over 9,500yen above buying rail travel without the Pass. Each. That's over 19,000yen in total. In my view jessieh would be better off spending that money on nicer hotels)