Generally speaking, when a lawyer files a motion, the opponent can file a response, and the first lawyer can file a reply. Occasionally, a court may allow a reply to the reply, called a surreply. Of course, you can tell what sometimes happens next…. From United States ex rel. Hockett v. Columbia/HCA Healthcare (D.D.C. 2007) (paragraph breaks added):

Relator filed a Motion seeking leave to file a surreply to HCA’s motion to dismiss …. HCA filed an opposition, equal in length to the surreply at issue, which stated no reason for rejecting the surreply, other than it being untimely and unnecessary. That opposition brief was really just a rehash of HCA’s argument in support of its motion to dismiss, and thus was in effect a sur-surreply, to which relator, for some reason, saw fit to reply, leaving this Court as the owner of what may be the world’s first sur-sur-surreply, a position in which no Court should ever find itself…
Since all of these responses argued the merits just as much as they argued why certain papers should or should not be accepted, the Court is left with something it never thought it would see, a sur-sursur-surreply (hereinafter, “reply4”). All of these papers, particularly the reply4, add very little that is new, and do not respond to any improper argument. We are now several steps removed from a substantive motion, and are faced only with filings about filings.
Eventually we reach a point where all this metapleading must stop, and this is that point.

Curiously, a quick search through Bloomberg Law dockets for references to “sursursursurreply” finds several authorizations for such pleadings, all from Judge P. Kevin Castel in federal district court in Manhattan — and generally with notations such as “I trust that will do it,” “that should do it for a while,” and “let me know if this is not adequate.” I wonder whether those ostensible permissions to file such briefs are actually subtle ways of telling lawyers to cut it out.

Thanks to reader Sai for the pointer.