Here is the definition of aperiodic tiling on Wikipedia.
"A tiling is called aperiodic if its hull contains only non-periodic tilings. The hull of a tiling $T\subseteq\mathbb{R}^d$ contains all translates T+x of T, together with all tilings that can be approximated by translates of T. Formally this is the closure of the set $\{T+x\ |\ x\in\mathbb{R}^d\}$ in the local topology. In the local topology (resp. the corresponding metric) two tilings are $\varepsilon$-close if they agree in a ball of radius $\frac{1}{\varepsilon}$. around the origin (possibly after shifting one of the tilings by an amount less than $\varepsilon$)."
Okay. So, wouldn't this be a valid construction of an aperiodic tiling? Take a squarefree word on $n$ letters (that's infinite in both directions). Take a square grid. For each of the $n$ letters choose a vertical, and a horizontal nub. The vertical nub will stick out the top of a square. The horizontal nub will stick out of the right side. There will be a hole for a vertical nub in the bottom, and a hole for a horizontal nub in the left side of each square. We can choose a different kind of nub for each side, giving up to $n^4$ possible tiles. Then, we just arrange these according to the squarefree word. That is, let each square be centred at a lattice point. Each coordinate is the index of a letter in the squarefree word. Let this letter give its nub in the corresponding direction. If we translate this we can't have arbitrarily large repeating sections. This would imply arbitrarily long strings of repetitions in the squarefree word, which do not exist by definition.
If I'm right, this definition makes aperiodic tilings a lot less exciting to me than they've been made out to be. This one is just a glorified square grid. At the very least, in my book, a tiling that strongly doesn't repeat should have the property that we can't continuously deform its tiles, holding them in place, to obtain a periodic tiling.