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In this paper we mainly study the dynamical complexity of Birkhoff ergodicaverage under the simultaneous observation of any number of continuousfunctions. These results can be as generalizations of [6,35] etc. to studyBirkhorff ergodic average from one (or finite) observable function to anynumber of observable functions from the dimensional perspective.  For any topological dynamical system with $g-$almost product property anduniform separation property, we show that any {\it jointly-irregular set}(i.e.,the intersection of a series of $\phi-$irregular sets over several continuousfunctions) either is empty or carries full topological pressure. In particular,if further the system is not uniquely ergodic, then the {\itcompletely-irregular set}(i.e., intersection of all possible {\it nonempty$\phi-$irregular} sets) is nonempty(even forms a dense $G_\delta$ set) andcarries full topological pressure. Moreover, {\it irregular-mix-regular sets}(i.e., intersection of some $ \phi-$irregular sets and $ \varphi-$regular sets)are discussed.  Similarly, the above results are suitable for the case of BS-dimension.  As consequences, these results are suitable for any system such as shifts offinite type or uniformly hyperbolic diffeomorphisms, time-1 map of uniformlyhyperbolic flows, repellers, $\beta-$shifts etc..

Topological Pressure for the Completely Irregular Set of Birkhoff Averages