On September 19, 2018 VMware officially closed the general support for VMware vSphere 5.5 and customers running version 5.5 should plan an upgrade of their infrastructure asap.
Although there is the option to purchase an extended support for up to two years for those that are still running vSphere 5.5, VMware strongly recommends the upgrade to version 6.5.
Currently you have the option to perform a direct upgrade from version 5.5 to 6.5 since the upgrade to version 6.7 is only supported from version 6.0 or later. Upgrading to version 6.5 not only is the recommended action, but administrators can take benefit of the new interesting features provided.
Upgrade to version 6.0 is not recommended due to the upcoming expiring support (March 12, 2020) and some technical issues. VMware vSphere 6.5 is more stable and the general support is available until November 15, 2021.
Running a not supported version in your environment means that patches and bug fixes won't be released any longer exposing your infrastructure to possible services disruption due to technical reasons or possible unauthorized accesses to your data due to bug such as Spectre or L1TF.
Before upgrading your infrastructure, consult the Lifecycle Product Matrix to know the support expiration dates.