We report the first experimental observation of extreme multistability in a controlled laboratory investigation. Extreme multistability arises when infinitely many attractors coexist for the same set of system parameters. The behavior was predicted earlier on theoretical grounds, supported by numerical studies of models of two coupled identical or nearly identical systems. We construct and couple two analog circuits based on a modified coupled Rössler system and demonstrate the occurrence of extreme multistability through a controlled switching to different attractor states purely through a change in initial conditions for a fixed set of system parameters. Numerical studies of the coupled model equations are in agreement with our experimental findings.