We present a sample of 1165 extreme emission-line galaxies (EELGs) at 4 < z < 9 selected using James Webb Space Telescope (JWST) NIRCam photometry in the Cosmic Evolution Early Release Science (CEERS) program. We use a simple method to photometrically identify EELGs with Hβ + [O iii] (combined) or Hα emission of observed-frame equivalent width (EW) > 5000 Å. JWST/NIRSpec spectroscopic observations of a subset (34) of the photometrically selected EELGs validate our selection method: All spectroscopically observed EELGs confirm our photometric identification of extreme emission, including some cases where the spectral-energy-distribution-derived photometric redshifts are incorrect. We find that the medium-band F410M filter in CEERS is particularly efficient at identifying EELGs, both in terms of including emission lines in the filter and in correctly identifying the continuum between Hβ + [O iii] and Hα in the neighboring broadband filters. We present examples of EELGs that could be incorrectly classified as ultrahigh redshift (z > 12) as a result of extreme Hβ + [O iii] emission blended across the reddest photometric filters. We compare the EELGs to the broader (subextreme) galaxy population in the same redshift range and find that they are consistent with being the bluer, high-EW tail of a broader population of emission-line galaxies. The highest-EW EELGs tend to have more compact emission-line sizes than continuum sizes, suggesting that active galactic nuclei are responsible for at least some of the most extreme EELGs. The photometrically inferred emission-line ratios are consistent with interstellar medium conditions with high ionization and moderately low metallicity, consistent with previous spectroscopic studies.