Why can't a primary cell be recharged?
In some cases, a primary cell can be recharged, e.g. Mn-alkaline cells can take a partial charge. However, there are a number of reasons primary cells cannot be fully recharged, and that secondary cells have to be specifically designed to allow recharging.


Gas evolution: once most of a cell is charged (though all material has not been converted to its original state), the electrolyte is electrolyzed (apologies for the pleonastic redundancy), usually evolving hydrogen and oxygen, causing the cell to rupture. Nickel-metal hydride cells, for example, get around that issue by incorporating a catalyst to recombine them, $\ce{2 H2 + O2 -&gt; 2 H2O + heat}$.
Physical change in the reactants: the standard "dry cell" (Leclanché cell) is made with an outer shell of zinc. As the cell discharges, parts of the shell are dissolved, eventually leaving holes. Reversing the current flow will cause zinc to redeposit, but not necessarily filling in the weakened areas. Even lead-acid cells designed for storage have limited lifetime because of the buildup of large, insoluble crystals of lead sulfate (sulfation, aka sulphation in GB), which may break off from the lead plates, preventing reversal of the chemical reaction during charging.
Physical damage to the insulator: metal dendrite may grow during charge-discharge cycling, piercing the dielectric between cells, even in well-designed secondary cells.