Users of seasonally adjusted time series produced by National Statistical Offices often expect consistency between high level series and their components. However, some component series may exhibit a pattern of meagre – negligible non-positive – values, as well as seasonal extremes. For example, agricultural series in countries with distinct seasons, such as New Zealand, may exhibit both features. Multiplicative seasonal adjustment typically utilizes a logarithmic transformation, but the meagre values can make this impossible, while the extremes engender huge distortions that can produce unacceptable seasonally adjusted figures. This feeds into consistency problems between the adjusted aggregate and its components. We propose a new method of extreme-value adjustment based on the maximum entropy principle, which results in replacement of the meagre values and extremes by optimal projections that utilize information from the available time series dynamics. The method is illustrated using New Zealand agricultural series.