Dense coding is the seminal example of how entanglement can boost quantum communication. By sharing an Einstein-Podolsky-Rosen (EPR) pair, dense coding allows one to transmit two bits of classical information while sending only a single qubit [1]. This doubling of the channel capacity is the largest allowed in quantum theory [2]. In this letter we show in both theory and experiment that same elementary resources, namely a shared EPR pair and qubit communication, are strictly more powerful than two classical bits in more general communication tasks. In contrast to dense coding experiments [3–8], we show that these advantages can be revealed using merely standard optical Bell state analysers [9, 10]. Our results reveal that the power of entanglement in enhancing quantum communications qualitatively goes beyond boosting channel capacities.