Can a website ONLY have a generic top level domain
From a technical standpoint, it's possible, and there are some examples:
These are country code TLDs, but the point is that DNS as a technology does support "dotless" domains. However, it appears that most generic TLDs are not allowed to have them due to ICANN polices. According to RFC 7085 such restrictions only apply to contracted TLDs, but it's not clear which TLDs fall under that restriction. Other sources suggest that the new TLDs would also be subject to such restrictions.
It is also worth noting that even if you can technically set up a "dotless" domain name (per depquid's answer), it might cost you a pretty penny.
ICANN just auctioned off the rights to control a bunch of new top level domains. The cost of owning a top level domain is $185,000. In addition to the money, the process also involved technical requirements for being able to operate that TLD, showing that the TLD would be put to good use, a public comment period, and fielding any objections.
Even then new top level domains may be restricted from operating a dotless domain based on RFC and ICANN rules.