Is there also a leveling effect for weak acids in water? The leveling effect says that any acid stronger than the hydronium ion ($\ce{H3O+}$) will completely dissociate in water. Hence no differentiation of acid strength is possible between eg HBr and HCl in water.

I was wondering why no one ever talks about the leveling effect for acids weaker than water. Isn't it so that these acids would not dissociate at all and hence no differentiation is possible on this side either?

Also, what is the reasoning behind the fact that acids stronger than the hydronium ion will completely dissociate in water? Why can't it be an equilibrium (say if it's only slightly stronger than hydronium)?
To answer your first question, weak acids don't get levelled in water. Each solvent has their acid-base equilibrium window, and water has one that is between 0-14, meaning that any solvent with a pKa within that range can be distinguished in water. There are other solvents with a ore narrow acid-base discrimination Window, such as methanol can acid, with a range between -10 to about 3, which will be unable to discriminate between weak acids. The reasoning behind the acid-base discrimination window is a bit long winded (and maybe a little confusing), so I will try to break it down as much as I can. 

Say you have an acid in water, then the pKa + pKb = pKw 
This is true for any acid or base in water. If pKa&lt;0 or pKb&lt;0 then then most of the compound will be dissociated in water, as an acid or base, respectively. This is undesirable, especially if the pKa/pKb&lt;&lt;0. As you have noted in your comments, when the pKa of a compound is less than 2 units compared to water, then in water only 1% of it will remain in the undissociated form. If pKa>pKw, you will obtain a negative pKb which will behave more like a base in water. It follows that compounds with a 0&lt; pKa&lt; pKw  will fulfill the condition where both pKa and pKb> 0. Now guess what pKw is? 14. Hopefully this might explain to you why water is able to distinguish between weak acids with a pKa between 0 to 14.