A compilation of daily extreme temperatures recorded at the Fabra Observatory (Catalonia, NE Spain) since 1917 up to 2005 has permitted an exhaustive analysis of the fractal behaviour of the daily extreme temperature residuals, DTR, defined as the difference between the observed daily extreme temperature and the daily average value. The lacunarity characterises the lag distribution on the residual series for several thresholds. Hurst, H, and Hausdorff, Ha, exponents, together with the exponent beta of the decaying power law, describing the evolution of power spectral density with frequency, permit to characterise the persistence, antipersistence or randomness of the residual series. The self-affine character of DTR series is verified, and additionally, they are simulated by means of fractional Gaussian noise, fGn. The reconstruction theorem leads to the quantification of the complexity (correlation dimension, mu*, and Kolmogorov entropy, kappa.) and predictive instability (Lyapunov exponents, lambda, and Kaplan-Yorke dimension, D-KY) of the residual series. All fractal parameters are computed for consecutive and independent segments of 5-year lengths. This strategy permits to obtain a high enough number of fractal parameter samples to estimate time trends, including their statistical significance. Comparisons are made between results of predictive algorithms based on fGn models and an autoregressive autoregressive integrated moving average (ARIMA) process, with the latter leading to slightly better results than the former. Several dynamic atmospheric mechanisms and local effects, such as local topography and vicinity to the Mediterranean coast, are proposed to explain the complex and instable predictability of DTR series. The memory of the physical system (Kolmogorov entropy) would be attributed to the interaction with the Mediterranean Sea.