It's known that finding a Gröbner basis of a polynomial ideal has a worst-case space complexity of $O(2^{2^{c\cdot n}})$, where c is constant and n is the number of variables $k[x_1,\ldots,x_n]$.
However, in practice it seems that most ideals have a simple Gröbner basis.
Can anyone give some concrete examples of small generators whose ideal has a large Gröbner basis? How would I go about searching for such examples (besides a brute-force approach of trying random ideals)?