At least one common quantity I can think of has dimension with a non-integer exponent. The specific detectivity, $\text{D}^*$ is a common descriptor of photodiodes, and I'm sure one could make an analogous figure of merit for other types of sensor.
The unit of $\text{D}^*$ is the "Jones," which is equal to
$$\frac{cm \cdot \sqrt{Hz}}{W}$$
Watts in SI decompose into
$$Kg \cdot m^2 \cdot s^{-3}$$
which makes one Jones equal to:
$$\frac{s^{2.5}}{Kg \cdot m} \times 10^{-2}$$
with dimension:
$$\text{time}^{2.5} \text{mass}^{-1} \text{length}^{-1}$$
There are other quantities related to $\text{D}^*$, such as Noise Equivalent Power (NEP), which come up a lot in radiometry. Basically, any measurement which is normalized by frequency bandwidth will end up with that $\sqrt{Hz}$ in the units.
There is really just one underlying principle here: Any equation we write down should not depend on any arbitrary choices we made in order to define the quantities. All the examples you can discuss can be understood in this principle.
Can't add a vector and a scalar. Well, of course a vector is three numbers and a scalar is one number, so, for example, $\mathbf{v} + v$, , where $v$ is a speed, i.e. a scalar with units of velocity, doesn't even make mathematical sense. But we could add imagine adding one component of a vector to a scalar, i.e. $v_z + v$. But, this quantity shouldn't appear in a fundamental law of physics because our choice of what axis to call the $z$ axis is completely arbitrary, and if we made a different choice our equations would look different. But this is situation-dependent. For example, if we were discussing physics in a background uniform gravitational field, then we can use a convention where $z$ points along the direction of gravitational field. This is not arbitrary because the gravitational field sets a preferred direction. By declaring that we are going to call that particular direction the "z-direction'', it makes sense that any equations we then write down will only hold for that particular choice of $z$-axis. That is why the equation for the gravitational potential energy in a gravitational field, $U = mgz$, is valid even though $z$ is a component of the displacement vector $\textbf{r}$. However, you can still translate this equation into one that is valid for any choice of axes, namely $U = m \textbf{g} \cdot \textbf{r}$, where $\textbf{g}$ is the gravitational field vector.
Can't add numbers with different units. The point is that we normally work in units of physics which are chosen completely arbitrarily. If time $t$ is measured in seconds and position $x$ is measured in meters, then it
makes no sense to write down an equation involving $x+t$ because this equation would depend on our definition of "second" and "meter", and there is no reason why the laws of physics should depend on the second being defined to be 9,192,631,770 times the period of some radiation mode of a cesium atom. But, if we choose to work in natural units, then this is not an arbitrary choice because, as the name suggests, natural units are uniquely determined given fundamental constants of physics. In natural units, there is nothing wrong with writing an equation involving $x+t$, because we remember that we have made a special choice of units, and the equation will hold only in those units.
Of course, any equation that you can write in natural units can still be translated into arbitrary units. Take Einstein's famous mass-energy equivalence. In natural units ($c=1$) it states that $E = m$. Obviously, in arbitrary units, this is a bad equation because if $E$ is measured in Joules, and $m$ is measured in kg, then it would depend on the definitions of Joules and kg. But that's fine, because this equation only holds in natural units. Its translation into arbitrary units is $E = mc^2$, and the units now match up.
Can't add covariant 4-vectors and contravariant 4-vectors. Again, this is because in special relativity, in order to write down components of vectors we have to make an arbitrary choice of coordinate directions in space-time. Equations we write down in special relativity shouldn't depend on this choice, and this prevents us from adding covariant 4-vectors and contravariant 4-vectors because they transform differently when you change coordinate directions in space-time.
Can't add things in different vector spaces. This is just because, if $\mathbf{v}$ is in one vector space and $\mathbf{w}$ is in a totally different vector space, then you wrote an equation involving $\mathbf{v} + \mathbf{w}$ then it would depend on how you relate the bases between the two vector spaces, which -- given that they are totally different spaces -- there is no way to do non-arbitrarily.
Best Answer
The key difference is the $ \frac{1}{4\pi\epsilon_0} $, with $ \epsilon_0 $ in the SI formulation of charge, being vaccuum permittivity with units $ (charge)^2(time)^2 (mass)^{−1}(length)^{−3} $. This satisfies the unit cancellation, and in the SI system makes the electric constant $ \mu_0 $ and $ \epsilon_0 $ now derived units. (See Vacuum Permittivity or SI Unit Redefinition)
- Wikipedia: Gaussian Units
Yes, I would argue that 'fundamental quantities' are indeed arbitrary, as are many of our choices, such as base-10 number systems. This is illustrated well on the Golden Record we put on voyager spacecraft, for decoding by other intelligent life; we show how fast to spin the record by relating time units in the fundamental transition of the hydrogen atom:
I'd then add that we have tried to make them as least-arbitrary (to us) as possible, but there's no reason that some other intellegence would have different 'fundamental unit' definitions and scalings, or whatever 'arbitrary' units they came up with. We could use $ (time)^{-1} $ or 'period' as our fundamental timing unit, and change all the other derived units to follow, if we wanted.