Using a submillimeter-sized YIG (yttrium-iron-garnet) sphere mounted in a magnetic-field-focusing cavity, we demonstrate an ultrahigh cooperativity of 105 between magnon and photon modes at millikelvin temperatures and microwave frequencies. The cavity is designed to act as a magnetic dipole by using a novel multiple-post approach, effectively focusing the cavity magnetic field within the YIG crystal with a filling factor of 3%. Coupling strength (normal-mode splitting) of 2 GHz (equivalent to 76 cavity linewidths or 0.3 Hz per spin) is achieved for a bright cavity mode that constitutes about 10% of the photon energy and shows that ultrastrong coupling is possible in spin systems at microwave frequencies. With straightforward optimizations we demonstrate that this system has the potential to reach cooperativities of 107, corresponding to a normal-mode splitting of 5.2 GHz and a coupling per spin approaching 1 Hz. We also observe a three-mode strong-coupling regime between a dark cavity mode and a magnon-mode doublet pair, where the photon-magnon and magnon-magnon couplings (normal-mode splittings) are 143 and 12.5 MHz, respectively, with a HWHM bandwidth of about 0.5 MHz.