The electron self-energy for long-range Coulomb interactions plays a crucial role in understanding the many-body physics of interacting electron systems (e.g., in metals and semiconductors) and has been studied extensively for decades. In fact, it is among the oldest and the most-investigated many-body problems in physics. However, there is a lack of an analytical expression for the self-energy Re Sigma((R)) (epsilon, T) when energy epsilon and temperature k(B)T are arbitrary with respect to each other (while both being still small compared with the Fermi energy). We revisit this problem and calculate analytically the self-energy on the mass shell for a two-dimensional electron system with Coulomb interactions in the high density limit r(s)<< 1, for temperature r(s)(3/2) << k(B)T/E-F << r(s) and energy r(s)(3/2) << vertical bar epsilon vertical bar << r(s). We provide the exact high-density analytical expressions for the real and imaginary parts of the electron self-energy with arbitrary value of epsilon/KBT, to the leading order in the dimensionless Coulomb coupling constant r(s), and to several higher than leading orders in k(B)T/r(s)E(F) and epsilon/r(s)E(F) . We also obtain the asymptotic behavior of the self-energy in the regimes vertical bar epsilon vertical bar << k(B)T and vertical bar epsilon vertical bar >> k(B)T. The higher-order terms have subtle and highly nontrivial compound logarithmic contributions from both epsilon and T, explaining why they have never before been calculated in spite of the importance of the subject matter.