Are there any quantities in the physical world that are inherently rational/algebraic? Whenever we measure something, it is usually inexact.  For example, the mass of a baseball will never be measured exactly on a scale in any unit of measurement besides "mass in baseballs that are currently being measured" and rational multiples thereof.
Are there any non-arbitrary physical quantities that are "inherently" rational?  That is, quantities that can be expressed "exactly" in a non-trivial, fundamental unit of measurement?
As far as I know, there is total charge, because charge can always be represented as an integer multiple of the charge of an electron.  And to an extent, photons/light can be described this way too.  Are there any others?
Also, are there any inherently "algebraic" quantities that are not inherently rational?  That is, quantities that can be expressed closed-form as the solution to an algebraic equation in a non-trivial unit of measurement, but not as an "exact" ratio to that unit of measurement?
 A: Inherently rational quantities are rare, and inherently algebraic quantities are rarer still. Real numbers are defined by a limit, so you need a large-system of some kind, one of whose dimensionless properties converges to a rational or algebraic number. This occurs in certain thermodynamic systems, and the quantum hall conductivity plateaus are a nontrivial quantum mechanical example.
The critical exponents of phase transitions provide a separate example in purely classical thermodynamics. These quantities have nothing directly to do with quantum mechanics, and they are discrete. In two dimensions, the exponents in two dimensions are rational for the Potts model, which is not free, but is a nontrivial conformal field theory. There are physical realizations of these systems, and the critical exponents can be measured in these. The exponents of physical systems are exactly determined by the models, and this is a statement of universality.
So if you take a two-dimensional surface which is magnetized, and look at zero external field, and vary the temperature near the Curie point, the exponent of the spontaneous magnetization vs. the distance from the critical temperature is 1/8.
A: 
Are there any non-arbitrary physical quantities that are "inherently" rational?

Absolutely! The conductivity plateaus in the integer and fractional quantum hall effects correspond to conductivities:
$$ \sigma = \nu e^2/h $$
The $\nu$ take on values which are integers (obviously) for the IQHE and fractions for the FQHE. So if you fix the "non-trivial, fundamental unit of measurement", as you put it, for conductance to be $e^2/h$ then the hall conductivities for the IQHE and FQHE are "inherently rational".
The energy levels of the Hydrogen atom also satisfy these criteria. The energy of the $n^\textrm{th}$ orbital is given by:
$$ E_n = - \frac{E_r}{n^2} $$
If you take the Rydberg energy $E_r$ to be a "fundamental unit of measurement" then $E_n$ again satisfy the criteria.
Some might argue that $E_r$ is not fundamental enough. However, the quantum of hall conductance $e^2/h$ certainly is as fundamental as it gets. There are plenty of other examples such as, as mentioned by you and others, charge and spin. I'd like to add magnetic flux to that list. Again, from the quantum hall effect, where $p$ quanta of magnetic flux $\phi_0$ pair up with $q$ electrons to give you a composite fermion with an anyonic exchange phase $pq$.
In general, (almost?) any observable that corresponds to a topological invariant has eigenvalues which are integer or rational.

Also, are there any inherently "algebraic" quantities that are not inherently rational?

Interestingly, that appears to be a harder question to answer. So I'll let someone handle this part ;-)
A: 
Blockquote Also, are there any inherently "algebraic" quantities that are not inherently rational?

When you say algebraic, do you mean quantities that can be organized as a Group? In my view, we can make measurements only up to an isomorphism which is imposed by your measuring device, so perhaps this answers your second question. 
