Repeating graphs whose lattice-link edges connect all the points of the square or of the triangular lattice can form the boundaries of symmetrical monohedral tessellations (tilings), termed lattice labyrinths. Requiring all lattice points to be connected implies that the prototiles of the tessellations are slim polyominoes and polyiamonds made up of corridors just one square or triangle wide, enclosing no lattice points and having boundaries of maximum length given their area. There is no upper bound to the area of the prototiles of the several infinitely populous families of lattice labyrinth tessellations. Lower-order examples invite Escherization and are a fertile source of logos. Despite being homeomorphic to the chessboard, honeycomb or other simple tilings, higher-order labyrinths are beguilingly intricate and the maximized boundary length and interpenetration of the prototiles may allow technical applications.