I want a name that creates a reaction from the public. At the beach they have a seafood place called Dirty Dicks. When I left the beach that name stuck with me. How far can I go to having a profane name to my store? Are there any limits on what I can call my store?
So as per recent Supreme Court decision in Matal v. Tam, the phohibitation of the registration of trademarks that may "disparage" persons, institutions, beliefs, or national symbols was ruled Unconstititutional. At issue, Tam, the bassist and founder of an all Asian-American band that was named after a slur for Asians (the linked article contains the Band Name, but to avoid a possible rule violation, I will not use the actual term). Tam was unable to register the trademark for his band due to the term being a slur, despite the bands intention to use the term in a re-appropriated manner.
SCOTUS held that the act was unconstitutional and that Trademark protection did not constitute government speech beyond that the specific mark was registered with the government and thus protected. The decision found 9-0 in favor of Tam. (The case gained some fame as it immediately ended litigation on the name of Washington D.C. NFL football team as well).
The other famous case is that of Jacobellis v. Ohio which held obsecenity is not protected speech, but that obscenity is to be determined by the most local levels of government. After all, what is Not Obscene to Los Angeles could be highly obscene to the Amish in Pennsylvania (Basically, no opinion was agreed too, but this was concurred with and every Justice held a concurring opinion). This case is famous for Justice Potter Stewart's concurring opinion, in which he stated obscenity should be limited to only Hard Core Porn, but famously did not offer a legal definition of that term beyond "I know it when I see it [and the motion picture involved in this case is not that]." Basically, check your local limits for guidance, as there is no federal or state law to help discuss this.
Finally, as a small aside, the specific name of the establishment in the OP may be on it's face a little naughty, but "Dick" is a diminutive name for "Richard" and this gets all sorts of playful puns on it's other vernacular uses. For example, in one comic featuring Batman's young ward Robin (Dick Grayson), who was admonishing other heroes for not protecting their secret identity, one character comments to another that "Robin is being such a -" only for the second character to interrupt and remind the first "No real names."
Another example is the stage name of a famous parody artist who sepcializes in singing modern pop songs in the style of a Vegas Crooner or Lounge Singer, Richard Cheese. The pun relies on knowing this nickname.
For one that is not punning on the body part, The Red Green Show once had a sketch where Red Green (the character's name) was talking with the town's notorious liar, Hap, who in this particular sketch dropped the time he went whaling and chased after the "Great White Whale Moby Richard." Red asks if he meant the titular whale from the novel Moby Dick, to which Hap responds, "I didn't know him that well, Red."