Lossy Sorting (Implement Dropsort)

APL, 9 bytes

⊢(/⍨)⊢=⌈\

This is a monadic function train with diagram:

┌─┼───┐  
⊢ ⍨ ┌─┼─┐
┌─┘ ⊢ = \
/     ┌─┘
      ⌈  

The non-train version is

{⍵/⍨⍵=⌈\⍵}

This basically checks if each element is equal to the running maximum.

Note that Martin Büttner's J solution is the same length as this and was posted first.

J, 10 9 bytes

#~(=>./\)

Working version of my CJam idea (in fewer bytes). E.g.:

   f =: #~(=>./\)
   f 10 10 10 9 10
10 10 10 10
   f 1 2 5 4 3 7
1 2 5 7

Explanation

First, we get the maximum of each prefix, with:

    >./\

(Here, >. is the maximum operator, / folds that operator onto a list, and \ gets all the prefixes of the input.)

Then we compare the initial list with those maxima for equality:

  (=>./\)

And finally, we select all elements where this list of boolean results gave a 1:

#~(=>./\)

Haskell, 28

foldr(\x l->x:filter(x<)l)[] 

An anonymous function. Call it like

foldr(\x l->x:filter(x<)l)[] [-7, -8, -5, 0, -1, 1] 
[-7,-5,0,1]

Equivalent to the recursion

f[]=[]
f(x:l)=x:filter(x<)(f l)

Translated iteratively, we iterate over the elements, and for each one we see, we remove the ones smaller than it from the remainder of the list that we're iterating over. Thanks to Antisthenes for a byte saved with (x<).