We use cosmic radiation records (neutron monitor and the cosmogenic radionuclides, 10Be and 14C) as a proxy to compare the solar activity during the extended solar minimum 2006–2009, with that during the Grand Solar Minima and Maxima that occurred between 1391 and 2010. The inferred cosmic ray intensities during the Spoerer, Maunder, and Dalton Grand Minima were significantly greater than those during 2006–2009. The onset phases of the three Grand Minima extended over between two and five Schwabe (sunspot) cycles, the cosmic ray intensity at the Schwabe minima increasing from a value approximating that of 2006–2009, to substantially higher values later in the Grand Minimum. The minimum estimated strengths of the heliospheric magnetic field near Earth during the Grand Minima were 2.4 nT (Spoerer), <2.0 nT (Maunder), and 2.6 nT (Dalton), compared to 3.9 nT in 2009. We conclude that the periods of highest solar activity during the Maunder Minimum approximated those near the sunspot minima between 1954 and 1996. The average ratio of the maximum to minimum estimated HMF in the six Schwabe cycles in the Maunder Minimum is 1.54 (range 1.30–1.85) compared to 1.52 (1.31–1.63) for the modern epoch suggesting similar operation of the solar dynamo in both intervals. The onset phase of the Maunder Minimum extending over five Schwabe cycles, and the large increase in cosmic ray flux (and decrease in estimated heliospheric magnetic field), leads us to speculate that the magnetohydrodynamic amplification in the solar dynamo exhibits a relaxation time well in excess of the 11 year period of the Schwabe cycle.