Exploring the xorspace
MATL, 11 bytes

t"G!Z~Ghu]n


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The last test case doesn't run in the online interpreter because of memory limitations, but runs offline in less than 2 seconds on a modern computer.

Explanation

For input of size n, this does the following:


Initiallize result to input.
Repeat n times:


Apply bitwise XOR to all pair of entries from current result and input.
Attach input values to the result.
Deduplicate.

The output is the number of elements of the final result.


Commented code.

t      % Implicit input: row vector. Duplicate
"      % For each (i.e. do as many times as the input size)
  G!   %   Push input as a column vector
  Z~   %   Bitwise XOR with broadcast, i.e. for all pairs of entries of the
       %   two arguments. The first argument is the accumulated result
       %   from the previous iteration, the second is the input vector
  G    %   Push input again
  h    %   Postpend
  u    %   Unique values. Gives a row vector
]      % End
n      % Number of entries. Implicitly display


Example

The intermediate results (steps 2.1 and 2.3) for input [256 259 3] are:

First iteration: [256 259 3] with [256 259 3]: computing all pairs of bitwise-XOR gives the matrix

  0   3 259
  3   0 256
259 256   0


Attaching [256 259 3] and deduplicating

0 3 259 256


Second iteration: current result [0 3 259 256] with [256 259 3]. After deduplicating this again gives 

0 3 259 256


Third iteration: again

0 3 259 256


So the output is 4 (number of entries of result).