Tropical geometry can be described as "algebraic geometry" over the semifield $\mathbb{T}$ of tropical numbers. As a set, $\mathbb{T}=\mathbb{R}\cup \{ -\infty\}$; this is endowed with addition being given by the (usual) maximum of real numbers and multiplication by the (usual) sum of real numbers. Recently there has been a lot of research on this kind of geometry. The obvious question is now: why are people interested in this?

There are three possible motivations I am aware of:

(1) "Polynomial" equations over $\mathbb{T}$ can be interpreted as linear equalities and inequalities over the classical reals. So people working in linear optimization and similar areas can use tropical geometry as an alternative, more algebraic approach that might lead to new methods and insights.

(2) In quantization jargon, one can interpret $\mathbb{T}$ as a "classical limit" of semifields which are all isomorphic to $\mathbb{R}^{\ge 0}$ with the usual addition and multiplication. More precisely, for any $q>0$ set $\mathbb{T}_q=\mathbb{R}\cup\{ -\infty \}$ and consider the bijection $\log_q:\mathbb{R}^{\ge 0}\to\mathbb{T}_q$ (setting $\log_q(0)=-\infty$). Pushing forward the usual semifield structure on $\mathbb{R}^{\ge 0}$ along $\log_q$, we get a semifield structure on $\mathbb{T}_q$. Then it is easy to check that the semifield $\mathbb{T}$ is, in the obvious sense, the limit of $\mathbb{T}_q$ as $q\to 0$. So if one likes to think in these terms (I do not, but that shall not bother me for now), then real algebraic geometry appears (with a grain of salt) as the quantum version of tropical geometry, which in turn gives tropical geometry an important rôle.

(3) This is only a post hoc justification. Some problems of classical algebraic geometry, mainly in enumerative geometry, have been solved by methods of tropical geometry. The usual strategy is as follows: we have a question in algebraic geometry, to which the answer is supposed to be an integer (say). We then set up an analogous question in tropical geometry, prove that the answers to the two questions agree, and then work on the tropical question, which is usually much simpler to answer.

Besides these three arguments, do you have any other motivations for studying tropical geometry?