Edit: Ok, I googled "complicated unknot" and found a paper "Hard Unknots and Collapsing Tangles" by Louis H. Kauffman and Sofia Lambropoulou and a thesis "Interactive Topological Drawing" by Robert Glenn Scharein. I went through both and entered the unknots they give into SnapPea. In all cases SnapPea says that the volume is zero and, futhermore, reports that the fundamental group is $\mathbb{Z}$ (one generator, no relators). The unknots they discuss include the Goeritz unknot, Freedman's unknot, and several unknots that require increasing the complexity of the diagram before decreasing. (ie via Reidemeister moves). Another knot that SnapPea handled (~55 crossings) was the one on page 135 of the thesis, which is claimed to defeat KnotPlot.
Before you try to fool SnapPea, remember that you'll almost certainly have to go above 16 (17?) crossings to do so - see http://www.springerlink.com/content/y10185316280vpu4/ for the tale of the tabulation of knots by Hoste and Weeks and, independently, Thistlethwaite. Here is a nice quote: "...our methods for nonalternating knots are not algorithmic. Instead we simply employ a collection of methods that work for $N \leq 16$."