Tile is designed to HAVE grout lines. The reason is that not all tile is square.
also, not EVERY tile is exactly the same thickness. There is an accepted warp page allowed in the tile industry as well.
That is why your Travertine is tight in some places and not in others. A MINIMUM
recommended grout line for ANY tile is 1/16". I doubt you will be able to have a NO Grout install that you would be happy with.