What about a Hamiltonian path in a triangulation of an ngon? If not, how long is the longest path?
There are two planar triangulations on 14 vertices without hamiltonian paths. This is the smallest size. For sure these are wellknown. Those answer the question for $n=3$. For $n=4$ the first examples appear at 12 vertices. Same for $n=5$. For $n=6$ one with 10 vertices. For $n=7$ one with 11 vertices. And so forth. 


It is a famous theorem of Whitney (1931^{*}) that a $4$connected planar triangulation has a Hamiltonian cycle. ^{ Example of nonHamiltonian triangulation from Joseph Malkevitch, obviously not $4$connected: removing $3$ vertices (surrounding one) disconnects. (A graph is $4$connected if it requires removal of $4$ vertices to disconnect it.) }
In response to the OP's query: ^{ A connected triangulated graph with no Hamiltonian path. } Perhaps the OP may be interested in this paper:



See the answer to this question. The magic word is "Kleetope". For more information than you imagined possible, see Guido Helden's Thesis. 

