Was Jesus finite or infinite? (human or God?) I've always heard it said that Jesus was "completely human" and "completely God".  

How could he be both human (finite by very nature) and God (infinite by very nature)?
It's one of those difficult things to comprehend. In fact, I think we may never be able to fully understand it. Jesus is God and has existed eternally as God (John 1:1, 18) but took on the additional nature of humanity during the incarnation. He fully possesses each nature, much in a sense that each member of the Trinity is fully God. His natures are also distinct much as the members of the Trinity are distinct. For instance, Jesus as man is not omniscient (Luke 2:52, Mark 13:32). Jesus as God cannot experience death. His two natures are completely distinct yet fully and completely unified in His single person.