Stretch your imagination a bit and imagine an almost empty universe with nothing in it but a very, very large asteroid (or an asteroid in our universe but in an almost empty patch so it's not affected too much by other celestial bodies).
I travel to this asteroid and I use some powerful drill and dig a tunnel to the center of mass of this asteroid.
Far away in another empty patch of the universe an atomic clock ticks exactly one second. Consider this the "proper" time interval.
The equation to calculate gravitational time dilation under the effect of a non-rotating object is
$$ \Delta t_{ 2 }= \Delta t_{ 1 } \sqrt { 1-\frac { 2Gm }{ rc^{ 2 } } } = \Delta t_{ 1 } \sqrt { 1-\frac { r_{ s } }{ r } } $$
where $r_s$ is the Schwarzschild radius, $r$ is my distance from the center of the center of mass, $\Delta t_2$ is the time interval experienced by the observer and $\Delta t_1$ is the proper time interval.
In order to try to solve the time dilation using this equation while I sit exactly on the center of mass I have to divide by 0 because my distance from the center is 0.
Since that would be mathematically undefined, how long has passed relative to an observer at the center of a non-rotating and not very massive object?
I've searched for other questions but they use other equations, consider black holes, talk about the Earth, or have been closed due to poor wording by the OP. My curiosity is what this can tell me about how one would experience time at the center of an asteroid.
