The performance of optical fibers is dependent on both the fiber design and the materials from which it is made. While much of the development over the past few decades has focused on fiber geometry and microstructuring, more recent analyses have shown clear benefits of addressing parasitic nonlinearities at the origins of their light–matter interactions. Reported here are results on intrinsically low Brillouin and thermo-optic core fibers, fabricated using modified chemical vapor deposition. Specifically, fibers in the Yb-doped Al2O3−P2O5−B2O3−SiO2 system are developed based on how each glass constituent affects the material parameters that enable both stimulated Brillouin scattering (SBS) and transverse mode instability (TMI). One fiber, developed to be very heavily doped, exhibited thermo-optic and Brillouin gain coefficients up to ∼3 dB and 6 dB below conventional laser fibers, respectively. A second fiber, designed to approximate a commercial double-clad laser fiber, which necessitated lower doping levels, was output power scaled to over 1 kW with an efficiency over 70% and no observed photodarkening under conventional testing. Design curves for the enabling material properties that drive TMI and SBS also are provided as functions of compositions as a tool for the community to further study and develop intrinsically low-nonlinearity fiber lasers.