The dynamic motion of a bubble rising through a shear-thickening fluid is numerically simulated. The simulations are carried out on a three-dimensional, dynamic, block-structured adaptive grid. The deforming bubble boundary is captured using the coupled level-set/volume-of-fluid (CLSVOF) method, which combines some of the advantages of the volume-of-fluid (VOF) method with those of the level-set (LS) method. The viscosity profile around the rising bubble is derived from numerical data, and it facilitates the determination of the shear-thickening effects as a function of the power-law exponent n. In general, as n increases, the bubble rise speed decreases and the amount of deformation of the bubble shape decreases. For different values of n, the effective viscosity and the associated effective physical dimensionless numbers are determined in order to quantify the shear-thickening effect of the continuous phase fluid on the bubble rise motion. One can analyze non-Newtonian bubble morphology by considering a non-Newtonian system with a specific effective viscosity.