This is about
US coins.
There are 6 kinds
currently in circulation: the 1-cent, 5-cent, 10-cent, 25-cent, 50-cent, and 1-dollar coins.
First's cost is twice itself
A 1-cent coin costs roughly 2 cents to produce (thanks @PlayerOne)
Second's of its name
A 5-cent coin is called a nickel and is made partly of nickel.
Third's tiniest of all
The 10-cent coin is the smallest of all.
Fourth's fourth
The 25-cent coin is called a quarter, and a quarter is a fourth or $\frac{1}{4}$.
Fifth's whole
Given the hint "skip the half", I think we're supposed to forget the 50-cent coin, so this is a whole dollar.
What is sixth?
A 2-dollar coin?