I wonder if one can tile the plane with an order-$n$ L-polyomino in a fundamentally irregular manner. I seek help in defining what should constitute "irregular."

An L-polyomino of order $n \ge 2$ is a
line of $n$ unit squares joined edge-to-edge, with one more square attached to the
$n$-th to form an *L*-shape.

I am wondering for which $n$ is there an order-$n$ L-polyomino that can tile the plane "irregularly." Two regularities I would like to avoid are: (1) within the tiling any rectangle (of any size) tiled by L-polyominoes; see (a) in the figure below, using order-$4$ polyominoes. (2) any "periodic crack," which I define as an infinite staircase, with a finite periodic series of steps up/down and right/left, that never includes a point strictly interior to an L-polyomino. A simple example is in (b) below, but in general steps up and down, right and left, of varying length, would constitute a periodic crack if repeated infinitely. Such a periodic crack partitions the tiling into to "halves"; it is in some sense an infinite "digital line" cleaving the tiling in two halves.

Finally, (c) below shows a partial tiling by order-$4$ polyominoes
which (so far) violates neither (1) nor (2).

Q1. Is there an irregular tiling by L-polyominoes under my definition of "irregular"?

Q2. Are there accepted definitions of what constitutes an irregular tiling, by one tile (a monohedral tiling)?

(*Addendum 1*). To respond to Zack's question, let us
insist that the tiles can only be rotated—no mirror reflections permitted.
(This may be not be standard notation...)

(*Addendum 2*). The chair tiling (from this link),
as per Anthony Quas, with my superimposed periodic staircase:

monotileproblem oreinsteinproblem and remains open - math.stackexchange.com/a/533959/29059. However, L-polyominos do tile periodically so are not a solution. $\endgroup$ – Dan Rust Jun 1 '14 at 9:17