We have searched for continuous gravitational wave (CGW) signals produced by individually resolvable, circular supermassive black hole binaries (SMBHBs) in the latest European Pulsar Timing Array (EPTA) data set, which consists of ultraprecise timing data on 41-ms pulsars. We develop frequentist and Bayesian detection algorithms to search both for monochromatic and frequency-evolving systems. None of the adopted algorithms show evidence for the presence of such a CGW signal, indicating that the data are best described by pulsar and radiometer noise only. Depending on the adopted detection algorithm, the 95 per cent upper limit on the sky-averaged strain amplitude lies in the range ... This limit varies by a factor of five, depending on the assumed source position and the most constraining limit is achieved towards the positions of the most sensitive pulsars in the timing array. The most robust upper limit - obtained via a full Bayesian analysis searching simultaneously over the signal and pulsar noise on the subset of ours six best pulsars -- is ... These limits, the most stringent to date at f < 10...nHz, exclude the presence of sub-centiparsec binaries with chirp mass ... out to a distance of about 25 Mpc, and with ... out to a distance of about 1Gpc (...). We show that state-of-the-art SMBHB population models predict <1 per cent probability of detecting a CGW with the current EPTA data set, consistent with the reported non-detection. We stress, however, that PTA limits on individual CGW have improved by almost an order of magnitude in the last five years. The continuing advances in pulsar timing data acquisition and analysis techniques will allow for strong astrophysical constraints on the population of nearby SMBHBs in the coming years. (ProQuest: ... denotes formulae/symbols omitted.)