We present a search for excited or exotic electrons decaying to an electron and a photon with high transverse momentum. An oppositely charged electron is produced in association with the excited electron, yielding a final state dielectron + photon signature. The discovery of excited electrons would be a first indication of lepton compositeness. We use $\sim$202 pb$^{-1}$ of data collected in {\mbox{$p\bar p$}}\ collisions at {\mbox{$\sqrt{s}$ =\ 1.96\ TeV }} with the Collider Detector at Fermilab during March 2001 through September 2003. The data are consistent with standard model expectations. Upper limits are set on the experimental cross-section $\sigma(\bar p p \rightarrow e e^* \rightarrow e e \gamma)$ at the 95\% confidence level in a contact-interaction model and a gauge-mediated interaction model. Limits are also presented as exclusion regions in the parameter space of the excited electron mass ($M_{e^*}$) and the compositeness energy scale ($\Lambda$). In the contact-interaction model, for which there are no previously published limits, we find $M_{e^*}<906$ GeV is excluded for $M_{e^*} = \Lambda$. In the gauge-mediated model, the exclusion region in the $M_{e^*}$ versus the phenomenological coupling $f/\Lambda$ parameter space is extended to $M_{e^*} < 430$ GeV for $f/\Lambda \sim 10^{-2}\ {\rm GeV^{-1}}$. In comparison, other experiments have excluded $M_{e^*} < 280$ GeV for $f/\Lambda \sim 10^{-2}\ {\rm GeV^{-1}}$. Posted to /cdf/pub/thesis/cdf7547_Gerberich_thesis.ps http://www-cdf.fnal.gov/thesis/cdf7547_Gerberich_thesis.ps Posted by hkg@fcdfwww