Universal gravity at small distance Could it be that there is simply a maximum gravitational force that two bodies of finite mass can exert on one another? This would occur at $r=0$, so maybe there is some really really really small $a$ in the universal gravity equation, making it 
$$F=G\frac{m_1m_2}{r^2+a}$$
If $a$ was small enough, it would only become apparent at extremely small distances. So as $r$ got smaller, the force would approach it's actual maximum which would be proportional to the two masses rather than just approaching infinity.
Feel free to tell me in layman's terms why that idea is no good, but I haven't had calculus in like 15 years so I doubt I will understand anything too complex.
 A: A note about the statement that a maximal gravitational force would occur at r=0: we can at least exclude the case for two distinct fermions with identical quantum numbers ,  by the Pauli Exclusion Principle.  
A: Folks are looking for non-Newtonian gravity.  The experimental gravity group at U. Washington have really been the leaders in the field over the past ten years; they have some nice review papers available for free.
Because of the way that short-range forces work in quantum mechanics,  we expect that a short-range gravitational interaction would produce a Yukawa potential,
\begin{align*}
\Phi = -\frac{GMm}{r}
\left( 
1  + \alpha e^{-r/\lambda}
\right).
\end{align*}
The first term is just the ordinary Newtonian potential.  In the second term, $\alpha$ sets the strength of the interaction and $\lambda$ is its length scale.  The constraints depend on both $\alpha$ and $\lambda$.  As of 2007, it was possible to rule out an interaction with $\alpha=1$ and $\lambda$ > 55 μm.
I expect that the existing data sets constraints on your $a$.  If you go to do arithmetic with it, you'll probably want to redefine your $a$ to have units of length rather than length$^2$.
