Is it possible to create a traditional clock solar system? Is it possible to create the following system of satellites?:


Place a satellite that orbits the some object X, once per day. Call it D.
Place a satellite that orbits D, once per hour. Call it H
Place a satellite that orbits H, once per minute. Call it M.
Place a satellite that orbits M, once per second. Call it S.


If possible, what are some values for the weights, rotational speed, and radii that would work?

Also, does the universe have a limit on how many layers of satellite systems it permits?
Is it possible to create the proposed system of satellites?

Yes (in theorey). Hierarchical multiple systems (like the one proposed in the OP) tend to be stable if the periods differ by $\sim 5$ or more and the orbits are near-circular. So, such a system could be stable. A slight problem may be the commensurability of the periods (the fact that their ratios are rational numbers), which implies orbital resonances. However, these resonances are rather weak (the lowest ratio is 1:24), but they may mess up the system in the very long term (a detailed analysis would be required to find out). In practice, it will be extremely hard to set-up such a system, even with advanced space engineering.

What are some values for the weights, rotational speed, and radii that would work?

Since this is a strongly hierarchical system, the masses $m$ and semi-major axes $r$ (=radius if the orbit is circular) can be worked out from Kepler's third law
$$
    \frac{2\pi}{T} = \Omega = \sqrt{\frac{G(m_1+m_2)}{r^3_{1-2}}}
$$
for each sub-system. Giving
\begin{align}
   \frac{4\pi^2}{1\mathrm{sec}^2} &amp;= \frac{G(m_S+m_M)}{r^3_{S-M}} \\
   \frac{4\pi^2}{1\mathrm{min}^2} &amp;= \frac{G(m_S+m_M+m_H)}{r^3_{SM-H}} \\
   \frac{4\pi^2}{1\mathrm{hour}^2} &amp;= \frac{G(m_S+m_M+m_H+m_D)}{r^3_{SMH-D}} \\
   \frac{4\pi^2}{1\mathrm{day}^2} &amp;= \frac{G(m_S+m_M+m_H+m_D+m_X)}{r^3_{SMHD-X}},
\end{align}
where $r_{ABC-D}$ denotes the distance (more precisely: the orbital semi-major axis) between object $D$ and the centre of mass of objects $A$, $B$, and $C$. Since these are 4 equations for 8 unknowns (4 masses and 4 distances), there is some freedom in the design of your 'clock'.

Does the universe have a limit on how many layers of satellite systems it permits?

So far, our formula only considered gravitational forces and neglected all else. Moreover, post-Newtonian effects (general relativity) have been ignored. As long as these assumptions remain valid (orbital speeds $\ll c$, orbital distances $\gtrsim\,$cm, solid objects with sizes $\ll$ distances so that tides are neglible, no electric or magnetic fields, vacuum), there is no limit. However, in practice, there is always some elctro-magnetic field and never perfect vacuum...