Is there anything we are aware of that can't be slowed by time dilation? While thinking about the ambitions of this post: Can radioactivity be slowed through time dilation?
I was asking myself, is there even anything that is dependent on time and can't be slowed through time dilation?
 A: No. It affects time on a fundamental level. All processes are affected.
Since the processes are occuring normally in the object’s own rest frame, and it is only the observer that perceives a different measure of time, clearly it must apply to everything without exception.
Consider two points in 4D spacetime. They represent an experimental box where different processes are performed: chemicals react, a spring unwinds, radioisotope decays, etc. Everything you can think of is represented.  One point is “before” and the other point is “after”.  In the experiment’s own frame there is no change in position and the interval between the two points is only in the time direction, of measure t1.
Some other observer will measure different values for the spacial separation and time t2.  Now does it even make sense to say that some processes will be measured at a time other than t2? That would mean they are no longer on that point: Observer 2 sees the spring unwind before the acid react, as opposed to observer 1 which sees them take the same time? That is not a different view of spacetime but a different reality.
The box (observer 1) is sitting there at rest minding its own business.  How can the presence of observer 2 (moving at relativistic speed relative to the box) somehow make the different processes act differently?
A: As pointed out in the comments in the answers to the referred question, Time Dilation is the property of the temporal interval between the end points of any process. It is not about the nature of the process, it is about the nature of the time interval itself. So whatever the process be, the time dilation is applicable equally. 
A: 
Is there anything we are aware of that can't be slowed by time dilation?

Yes. Black hole growth and the expansion of the universe. 
Time dilation is related to the coordinate speed of light, which varies from place to place. It varies in the room you're in, see this answer for some references. Also see this Wikipedia stub where you can read that "at the event horizon of a black hole the coordinate speed of light is zero". That's where gravitational time dilation goes infinite. But this doesn't stop the black hole growing. 
On top of that, you will be familiar with black hole singularities and the big bang singularity. They aren't quite the same, but they are related in that the changes you might notice when you moving away from a black hole through space are somewhat similar to the changes you might notice as the universe expands over time. And as you will know, the universe is expanding such that distant portions of it are moving apart by more than one light year per year. See Wikipedia: "Due to the expansion increasing as distances increase, the distance between two remote galaxies can increase at more than 3×108 m/s..."
The expanding universe is immune to any light-speed restriction. And when you wind the clock back to the early universe, you can reason that at some early time the energy density was like that of a black hole, and that time dilation was infinite like it is at the black-hole event horizon. And because time dilation and the coordinate speed of light are related, it means the expanding universe is immune to time dilation too. If it wasn't, we wouldn't be here. 
A: Time dilation is a concept of Minkowski geometry, i.e. the emergence of time dilation is a feature of the world geometry underlying Special Relativity. Therefore, your question boils down to the question what the fundamental geometric structure of the world really is. If the world geometry was truely Minkowskian, i.e. if Special Relativity was the final description of reality, time dilation would be fundamental (as outlined in previous answers).
As we know that the world geometry is not Minkowskian, and since we do not possess a complete and fundamental theory of physics (including a complete and fundamental theory of the world geometry), we do not really know in which regime the concept of time dilation applies.
A: The answer to your question is yes, Em waves. 


*

*Imagine a photon going through a tunnel through the center of mass, and another going around the mass. If time dilation would affect them, they would arrive at another far away point at the same time (saying that their paths' will cross again).

*It is because the one going through the center of mass would have a slower ticking clock (seem slower to us observers), and would travel shorter distance in 3D and the one going around would travel longer in 3D but would also have a faster ticking clock. So speed would be distance/time. smaller/smaller would be equal to bigger/bigger.

*But to an external observer, watching the two photons arrive at the same time, would mean that to our clock, they traveled different 3D distances in the  same amount of time, so their speeds can't be the same. And that is not possible.

*Shapiro delay shows you I believe that the photon around traveled in more time, than it should have. (although there is no experiment for testing another photon traveling through the center mass).
