In the first Derek Landy book, the protagonist has no powers; she gets by on being awesome and smart and having good allies. However, in later books she starts learning bending^H^H^H^H^H^H elemental magic. (I rather wish the author hadn't gone that way.)

Not a protagonist, but I very much appreciate that Diane Duane's later Wizard books develop Carmela as a nonwizard character who is formidable anyhow.