Why can Naruto use Rasengan more than Kakashi using Chidori? Shinobi using Rasengan can often use it more times in a battle as compared to a shinobi using Chidori. Kakashi on one hand, could only use Chidori upto a maximum of four times a day in Part 1. Naruto, however, even when he is using Shadow clones (meaning that he would have only a fraction of the chakra he normally has) still has no trouble using the Rasengan more than that and still have plenty of chakra. 

The Rasengan is almost equally, if not more, powerful than the Chidori. So the Rasengan should require the same or more amount of chakra than the Chidori.  Naruto is also said to be very inefficient in using jutsu. So even though he has more chakra than Kakashi, his Rasengan would be less efficient than Kakashi's Chidori. 
I do not believe that Naruto chakra levels are so vast, that even with the use of shadow clones and being inefficient while using jutsu, he can use it multiple times while still retaining plenty of chakra. 

So why can Naruto use the Rasengan more than Kakashi using the Chidori?
If I recall correctly, at one time, Kakashi commented that Naruto's chakra is about 4 times of what he has. With addition to Kyuubi's chakra, Naruto has about 100 times what Kakashi has. Suppose that Kakashi's chakra is 4, Chidori then costs 1 chakra for each use. Naruto has 4 times what Kakashi has so he has 16. Kage Bunshin actually costs small amount of chakra. The problem with Kage Bunshin is that it splits the chakra of the user which makes it appear as if it costs a lot.

But, when the jutsu is cancelled either by when the Bunshin gets hit, or the caster cancels it, the chakra comes back to the caster along with its experience. Suppose that Naruto doesn't cancel and re-cast Kage Bunshin, there won't be additional chakra cost for casting the jutsu.

Chidori cost = 1

Rasengan cost = 1

Kage Bunshin cost = 0.5 &lt;- assuming it cost lower than Rasengan

Kakashi Chakra = 4

Naruto Chakra = 16

Inefficient chakra usage modifier = 200% (was 125% but changed as per ThatOneGuys's comment)

Kakashi Max Chidori = 4 / 1 = 4

Naruto Max Rasengan = (16 - 0.5 * 200%) / (1 x 200%) = 15 / 2 = 7.5 -> 7 (rounded down)

Without using Kyuubi's chakra, Naruto can do about 2 times more Rasengan than Kakashi can use Chidori. With Kyuubi's chakra, Naruto would be able to do about 100 times more. Of course, since with the training Naruto improved his chakra control, the current Naruto can do more Rasengan than this simple calculation shows.

The reference about Naruto's and Kakashi's chakra pool size and the fact about Kage Bunshin splitting the user's chakra and returning it when the jutsu was cancelled is from the episode where Kakashi train Naruto to develop Rasen Shuriken.