How large could neutral Turkey's wartime navy become?

How large could the Turkish Navy become in World War II while Turkey still maintained neutrality?

A useful regional fleet was possible, but multiple capital ships or carriers would make Turkey too important for the belligerents to ignore.

Turkey could build a useful regional fleet without necessarily abandoning neutrality. The suggested difficulty begins when its strength approaches that of the powers associated with the Washington treaty system. More than one or two capital ships or carriers, or more than roughly five or six large cruisers, would make other states increasingly attentive to its allegiance. A fleet of four to six capital ships would put Turkey into a different strategic category. These are approximate political thresholds within the counterfactual, rather than treaty limits imposed on the proposed Turkish fleet.