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Logarithmic enumerative geometry is populated by a number of geometric/combinatorial constructions that appear over and over, in particular they always appear whenever talking about degenerations and gluing formulas. Through concrete examples, I will introduce some of these recurring characters: Artin fans, tropical models, logarithmic modifications, and fiber products in the category of fine and saturated logarithmic schemes. I will explain the role of combinatorial flatness and that logarithmic and ordinary fiber products need not coincide. For each log character, I will try to sketch how it appears in log Gromov–Witten theory.