  We numerically construct static and spherically symmetric electrically
charged black hole solutions in Einstein-Born-Infeld gravity with massive
dilaton. The numerical solutions show that the dilaton potential allows many
more black hole causal structures than the massless dilaton. We find that
depending on the black hole mass and charge and the dilaton mass the black
holes can have either one, two, or three horizons. The extremal solutions are
also found out. As an interesting peculiarity we note that there are extremal
black holes with an inner horizon and with triply degenerated horizon.
