Is there a research name for "systems that steadily grow in complexity as they are computed"?
Hypothesis could be :


The time is discrete, let $s_i$ be the state of your system at $t=i$.
Your system is deterministic and  its transition function $f$ is computable.
Its complexity $c : \mathbb{N} \to \mathbb{N}$ is the Kolmogorov complexity, $c(i)$ is the length of the shortest C++ program which prints $s_i$. 


Then you have a bound : $$c(i) \leq O^+(1) + c_0 + |f| + |i| = O^+(1) + |i| $$

($O^+(1)$ is a notation for a positive constant, |x| is the Kolmogorov complexity of x)

And more generally for any computable function $g$ : $$c(g(i)) \leq O^+(1) + c_0 + |f| +|g| + |i| = O^+(1) + |i| $$

The system complexity grows more slowly that any computable function ! 

Conclusion : 


You should look at systems which are not deterministic or whose transition function is not computable. Or use an other definition for complexity (like this one : Chaos Theory).
With Kolmogorov complexity, chaotic states are more complex than states with lifeforms defending themselves, but may be one can define a physical macroscopic Kolmogorov Complexity where the  unbreakable basic components for describe the world are quite big and their internal features are negligible. Then it becomes more complex to describe tangle of life and vacuum than homogeneous chaos.