The scalar structure function scaling exponent ζn is experimentally determined for n ≤10 in decaying, grid-generated wind-tunnel turbulence with a constant mean temperature gradient. The Reynolds number is varied over the range 150≤Rλ≤ 700 by using static and active grids. The results show that up to n = 10 the scaling exponent does not saturate although saturation is not precluded at higher orders. There appears to be no dependence of ζ n on Reynolds number and the values of ζ n are the same for the transverse (along the gradient) and the longitudinal (streamwise) structure functions. A compilation of previous work shows large variation in ζn, with a few results indicating saturation and most not. Reasons for the scatter are attributed to convergence problems at high orders, effects of flow or computational domain size causing clipping of large rare fluctuations, and differences in initial and boundary conditions.