What is the maximum covalency of oxygen?
We know the maximum covalent is at least 3 with such species as  $\ce{H3O+}$.  Also the pyrylium ion (https://en.m.wikipedia.org/wiki/Pyrylium_salt) has an oxygen covalency of 3 by forming a pi bond for the third bond.
Can we get 4-coordination?  As mentioned by @Rob, yes.   @Rob's comment states that basic beryllium acetate appears to have 4-coordinate, covalently bound oxygen.
And it check out.  Basic beryllium acetate, $\ce {Be4O(C2H3O2)6}$, has the four-coordinate oxygen in the center, then four beryllium atoms coordinated to this oxygen at the vertices of a tetrahedron, then an acetate group bridging each edge of the tetrahedron.  In addition to beryllium, zinc forms a similarly structured basic acetate, which can be obtained by heating normal zinc acetate (https://en.wikipedia.org/wiki/Zinc_acetate).
Might the central oxygen be mostly ionically bonded?  Magnesium and heavier alkaline earth metals, which form more strongly ionic bonds with oxygen, are not known to form such a basic acetate structure.  This corroborates the hypothesis that covalent bonding of the central, four-coordinate oxygen is involved.