In this paper, we discuss the problem of computing repeated squarings (exponentiations to a power of 2) in finite fields with polynomial basis. Repeated squarings have importance, especially, in elliptic curve cryptography where they are used in computing inversions in the field and scalar multiplications on Koblitz curves. We explore the problem specifically from the perspective of efficient implementation using field-programmable gate arrays (FPGAs) where the look-up table (LUT) structure helps to reduce both area and delay overheads. In fact, we show that the optimum construction depends on the size of the LUTs. We propose several repeated squarer architectures and demonstrate their practicability for FPGA-based implementations. Finally, we show that the proposed repeated squarers can offer significant speedups and even improve resistivity against side-channel attacks.