We design, fabricate, and characterize a multilayer nanophotonic structure that couples light from standard optical fiber to an integrated photonics chip with unprecedented efficiency. The structure comprises a multilayer waveguide array tapering into a single waveguide supporting only fundamental TE- and TM-like modes. Measurements reveal a record-setting fiber-to-chip coupling efficiency of ${98.3}\% \;{\pm}\;{0.3}\%$ per facet at a 1575 nm wavelength that remains greater than ${92.8}\% \;{\pm}\;{0.4}\%$ across the 1550-1600 nm wavelength range. This approach is tailorable to any material platform, fiber type, or operating wavelength and represents a significant step forward for the accessibility of integrated photonics.