There are 360 but your globe wont show all of them.
Definitive answer (sorry y'all): The number of lines of longitude is boundless. That is to say uncountably finite in number. The number of lines depicted on a world globe (model) is usually between 18 and 36, depending on mfr and build; but these are only representations for convenience. Any line of longitude is not a real thing, only an abstraction by which location is pinpointed and recorded with reference to its angular distance from another, fixed, reference longitude line on a spherical body or its globe representation..