A quantum algorithm solves computational tasks using fewer physical resources than the best-known classical algorithm. Of most interest are those for which an exponential reduction is achieved. The key example is the phase estimation algorithm, which provides the quantum speedup in Shor's factoring algorithm and quantum simulation algorithms. To date, fully quantum experiments of this type have demonstrated only the read-out stage of quantum algorithms, but not the steps in which input data is read in and processed to calculate the final quantum state. Indeed, knowing the answer beforehand was essential. We present a photonic demonstration of a full quantum algorithm—the iterative phase estimation algorithm (IPEA)—without knowing the answer in advance. This result suggests practical applications of the phase estimation algorithm, including quantum simulations and quantum metrology in the near term, and factoring in the long term.