Using the LLL algorithm and the second author's "ladder'' method, we find (conjectural) $\Z$-linear relations among polylogarithms of order up to 16 evaluated at powers of a single algebraic number. These relations are in accordance with theoretical predictions and are valid to an accuracy of 300 decimal digits, but we cannot prove them rigorously.
"A sixteenth-order polylogarithm ladder." Experiment. Math. 1 (1) 25 - 34, 1992.