The electromagnetic structure of the lightest hadrons, proton, pion, and kaon is studied by high-precision measurements of their form factors for the highest timelike momentum transfers of $|Q^2|=s=14.2$ and 17.4 GeV$^2$. Data taken with the CLEO-c detector at $\sqrt{s}=3.772$ and 4.170 GeV, with integrated luminosities of 805 and 586 pb$^{-1}$, respectively, have been used to study $e^+e^-$ annihilations into $\pi^+\pi^-$, $K^+K^-$, $p\bar{p}$. The dimensional counting rule prediction that at large $Q^2$ the quantity $Q^2F(Q^2)$ for pseudoscalar mesons is nearly constant, and should vary only weakly as the strong coupling constant $\alpha_S(Q^2)$ is confirmed for both pions and kaons. However, the measurements are in strong quantitative disagreement with the predictions of the existing quantum chromodynamics-based models. For protons, it is found that the timelike form factors continue to remain nearly twice as large as the corresponding spacelike form factors measured in electron elastic scattering, in significant violation of the expectation of their equality at large $Q^2$. Further, in contrast to pions and kaons, a significant difference is observed between the values of the corresponding quantity $|Q^4|G_M(|Q^2|)/\mu_p$ for protons at $|Q^2|=14.2$ and 17.4 GeV$^2$. The results suggest the constancy of $|Q^2|G_M(|Q^2|)/\mu_p$, instead, at these large $|Q^2|$.