Higher-order algorithms
there are a bunch of algorithms that are "truly 2nd order" although usually not explicitly described in these terms.  Whenever we have sub-linear time algorithms, implicit is some kind of oracle access to the input, i.e. really treating the input as a function.  

Examples:

(1) The Ellipsoid algorithms when working with a "separation oracle" (eg http://math.mit.edu/~vempala/18.433/L18.pdf )

(2) Submodular function minimization (eg http://people.commerce.ubc.ca/faculty/mccormick/sfmchap8a.pdf )

(3) The whole field of property testing is really of this form ( http://www.wisdom.weizmann.ac.il/~oded/test.html )

(4) Combinatorial auctions in the query model (eg  http://pluto.huji.ac.il/~blumrosen/papers/iter.pdf )