This is similar to what daw tries to do (as I understand it) but I hope I can explain the idea better and produce a better coverage at the same time.

Alice will use the following basic shape

 .#..
 ###.
 .###
 ..#.
 

and arrange lots of pieces like that in a pattern so that the don't touch each other. She doesn't have to lie a complete basic shape in one move, she just fills in spots that ought to be tiled in the large pattern.

 .#.###.
 ###.#.#
 .###.##
 #.#.#.#
 ##.###.
 The empty spots form infinite lines along some diagonals and in between you can always fit exactly one piece.
 

A little bit of counting now shows that

We need 4 empty fields for each 8 filled fields, giving us a coverage of 8/12=2/3

of the total area. In each turn Alice adds 10 new pieces and Bob can remove at most

8

So in the long run Alice can cover arbitrarily large parts of the plane with this pattern.

This is similar to what daw tries to do (as I understand it) but I hope I can explain the idea better and produce a better coverage at the same time.

Alice will use the following basic shape

 .#..
 ###.
 .###
 ..#.
 

and arrange lots of pieces like that in a pattern so that the don't touch each other. She doesn't have to lie a complete basic shape in one move, she just fills in spots that ought to be tiled in the large pattern.

 .#.###.
 ###.#.#
 .###.##
 #.#.#.#
 ##.###.

The empty spots form infinite lines along some diagonals and in between you can always fit exactly one piece.

A little bit of counting now shows that

We need 4 empty fields for each 8 filled fields, giving us a coverage of 8/12=2/3

of the total area. In each turn Alice adds 10 new pieces and Bob can remove at most

8

So in the long run Alice can cover arbitrarily large parts of the plane with this pattern.