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Fictitious play (FP) is one of the earliest and most fundamental learning procedures in games, where each agent independently chooses a best response to the empirical distribution of opponents' past actions by maximizing the expected payoff with respect to it. However, learning agents may not always be expected-utility maximizers: they may use nonlinear evaluation functionals to account for the uncertainty induced by the opponents, e.g., via risk measures. In this work, we study nonlinear fictitious play (NL-FP), in which each pure action is evaluated by a general, possibly nonlinear functional of the payoff distribution induced by the opponent's empirical belief. To analyze NL-FP, we first identify a single-crossing property (SCP) of evaluation functionals that preserves the classical geometry of FP in $2\times2$ games. Furthermore, we conduct a novel geometric analysis of the convergence behavior of NL-FP in $2\times2$ games, even when the SCP fails. Beyond $2\times2$ games, we show that the convergence of NL-FP in $2\times n$ ($n\ge3$) games may fail even for continuous-time NL-FP, when the two-action player uses conditional value-at-risk (CVaR) in a $2\times3$ game. In addition, we identify a finite-crossing condition that ensures convergence of the selected continuous-time dynamics and the discrete-time iterates in general $2\times n$ games, which also provides new insights into the global landscape of continuous-time classical FP in such games.