<p>In 2015, Cartwright (2016) showed that any 3-regular metric graph arises as the skeleton of a tropical plane curve with nodes allowed. They introduced the tropical crossing number of a metric graph as the minimum number of nodes required for that graph with the prescribed lengths. We introduce the tropical crossing number of a finite, non-metric graph, the minimum number of nodes required to achieve that graph with <i>any</i> lengths on its edges. We prove that tropical crossing numbers can be arbitrary, even with prescribed crossing numbers. More precisely, for any integers <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(0\le c\le d\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>c</mi> <mo>≤</mo> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation>, there exists a finite graph with crossing number <i>c</i> and tropical crossing number <i>d</i>. The same result holds for metric graphs, choosing an appropriate metric on the finite example. We then introduce and use computational methods to find the tropical crossing number of the smallest non-tropically planar graph, the lollipop graph of genus 3. We also show that our tropical crossing number can grow quadratically in the number of vertices of the graph.</p>