The US government currently spends significant resources managing the legacies of the Cold War, including 300 million liters of highly radioactive wastes stored in hundreds of tanks at the Hanford (WA) and Savannah River (SC) sites. The materials in these tanks consist of highly radioactive slurries and sludges at very high pH and salt concentrations. The solid particles primarily consist of aluminum hydroxides and oxyhydroxides (gibbsite and boehmite), although many other materials are present. These form complex aggregates that dramatically affect the rheology of the solutions and, therefore, efforts to recover and treat these wastes. In this paper, we have used a combination of transmission and cryo-transmission electron microscopy, dynamic light scattering, and X-ray and neutron small and ultrasmall-angle scattering to study the aggregation of synthetic nanoboehmite particles at pH 9 (approximately the point of zero charge) and 12, and sodium nitrate and calcium nitrate concentrations up to 1 m. Although the initial particles form individual rhombohedral platelets, once placed in solution they quickly form well-bonded stacks, primary aggregates, up to, similar to 1500 angstrom long. These are more prevalent at pH = 12. Addition of calcium nitrate or sodium nitrate has a similar effect as lowering pH, but approximately 100 times less calcium than sodium is needed to observe this effect. These aggregates have fractal dimension between 2.5 and 2.6 that are relatively unaffected by salt concentration for calcium nitrate at high pH. Larger aggregates (>similar to 4000 angstrom) are also formed, but their size distributions are discrete rather than continuous. The fractal dimensions of these aggregates are strongly pH-dependent, but only become dependent on solute at high concentrations.