What are the simplest turing-complete systems?
The Rule 110 cellular automaton (often simply Rule 110) is an elementary cellular automaton with interesting behavior on the boundary between stability and chaos. In this respect it is similar to Game of Life. Rule 110 is known to be Turing complete. This implies that, in principle, any calculation or computer program can be simulated using this automaton.

Another simple example is a Bitwise Cyclic Tag System:

The "program" is a sequence of instructions $0, 01, 11$ (i.e. $C = p_{n} p_{n-1} ... p_2 p_1; \; p_i \in (0|01|11)^+$)
; the "data" is a binary string (i.e. $w \in \{0,1\}^+$).

If $p_1$ is the "current instruction" and the data is $w = a_1 a_2 ... a_m$; you can put the program and the data side by side ($p_{n}...p_2 p_1 \leftrightarrow a_1 a_2 ...a_m$), and the "execution" of $p_1$ is:


if $a_1 = 1$ and $p_1 = x1$ then append $x$ to $w$
delete $a_1$ from $w$
rotate $C$ ($p_1$ is moved to the left of $C$ and $p_2$ becomes the current instruction)


The universality can be proved by simulating the more general cyclic-tag system.