here is the code for you. I use the cellfun to do this and I have comments after each line of the code. It can compute and store from fisrt - nth order of the self-multiplication of an arbitrary matrix m. If you have any question, feel free to ask.

function m_powerCell = power_store(m, n) %m is your input matrix, n is the highest power you want to reach

  n_mat = [1:n]; %set a vector for indexing each power result in cell

  n_cell = mat2cell(n_mat,1,ones(1,n)); %set a cell for each of the power

  m_powerCell = cellfun(@(x)power(m, x), n_cell, 'uni', 0);  %compute the power of the matrix

end

%this code will return a cell to you, each element is a matrix, you can
%read each of the matrix by m_powerCell{x}, x represents the xth order 

Use bsxfun again, but arrange the exponents (b) in a third dimension:

A = [1, 2 3; 4 5 6];
b = 1:10;
C = bsxfun(@power, A, permute(b(:), [2 3 1]));

This gives you a 3D array as result (2x3x10 in this case).

If exponents are consecutive values, the following code may be faster:

n = 10; %// compute powers with exponents 1, 2, ..., n
C = cumprod(repmat(A, [1 1 n]) ,3);

Like many MATLAB functions, the exp function operates element-wise when applied to arrays. For further details, please refer to the documentation.