We used the VLA to make deep images of two 7x7' fields at 8.44 GHz with 10" resolution. With an rms noise of 3.2 and 5.1 {micro}Jy, respectively, in the two fields, we compiled a catalog of 82 sources. The normalized differential 8.44 GHz counts are similar to those at 1.41 and 4.86 GHz. All show a similarly steep submillijansky slope, which is only somewhat flatter than that expected for a nonevolving Euclidean population ({gamma}=2.5). Microjansky radio sources at 4.86 GHz have been identified with faint blue galaxies (18<~V<~28 mag). We argue that their expected median redshift is about 0.5-0.75. Hence, cosmological evolution may be needed to explain the steep slope of the microjansky counts. The 8.44 GHz counts must converge with slope {gamma}<2.0 below S(8.44)~300 nJy, or they would exceed the available field galaxy counts down to V~28 mag, and they must permanently converge below S(8.44)~20 nJy, or their integrated sky brightness would distort the observed thermal cosmic background radiation spectrum at centimeter wavelengths. The estimated 31.5 GHz sky brightness from nanojansky to jansky levels is <36 {micro}K (3{sigma}). Even if weak radio sources cluster on scales of degrees as faint galaxies do, their anisotropic contribution to the COBE DMR experiment (with 7deg FWHM-beam) would not exceed ~1.2{micro}K.