Aims. We report the γ-ray activity from the intermediate BL Lac S5 0716+714 during observations acquired by the AGILE satellite in September and October 2007. These detections of activity were contemporaneous with a period of intense optical activity, which was monitored by GASP–WEBT. This simultaneous optical and γ-ray coverage allows us to study in detail the light curves, time lags, γ-ray photon spectrum, and Spectral Energy Distributions (SEDs) during different states of activity. Methods. AGILE observed the source with its two co-aligned imagers, the Gamma-Ray Imaging Detector (GRID) and the hard X-ray imager (Super-AGILE), which are sensitive to the 30 MeV–50 GeV and 18–60 keV energy ranges, respectively. Observations were completed in two different periods, the first between 2007 September 4–23, and the second between 2007 October 24–November 1. Results. Over the period 2007 September 7–12, AGILE detected γ-ray emission from the source at a significance level of 9.6-σ with an average flux (E > 100 MeV) of (97 ± 15) $\times$ 10-8 photons cm-2 s-1, which increased by a factor of at least four within three days. No emission was detected by Super-AGILE for the energy range 18–60 keV to a 3-σ upper limit of 10 mCrab in 335 ks. In October 2007, AGILE repointed toward S5 0716+714 following an intense optical flare, measuring an average flux of (47 ± 11) $\times$ 10-8 photons cm-2 s-1 at a significance level of 6.0-σ. Conclusions. The γ-ray flux of S5 0716+714 detected by AGILE is the highest ever detected for this blazar and one of the most intense γ-ray fluxes detected from a BL Lac object. The SED of mid-September appears to be consistent with the synchrotron self-Compton (SSC) emission model, but only by including two SSC components of different variabilities.