Is the repository too big overall? Ideally, Git repositories should be under 1 GiB, and (without special handling) they start to get unwieldy over 5 GiB. Big repositories take a long time to clone and repack, and take a lot of disk space.
Does the repository have too many references (branches and/or tags)? They all have to be transferred to the client for every fetch, even if your clone is up-to-date. Try to limit them to a few tens of thousands at most.
Does the repository include too many objects? The more objects, the longer it takes for Git to traverse the repository's history, for example when garbage-collecting.
Does the repository include gigantic blobs (files)? Git works best with small- to medium-sized files. It's OK to have a few files in the megabyte range, but they should generally be the exception.
Does the repository include many, many versions of large text files, each one slightly changed from the one before? Such files delta very well, so they might not cause your repository to grow alarmingly. But it is expensive for Git to reconstruct the full files and to diff them, which it needs to do internally for many operations.
Does the repository include gigantic trees (directories)? Every time a file is modified, Git has to create a new copy of every tree (i.e., every directory in the path) leading to the file. Huge trees make this expensive. Moreover, it is very expensive to traverse through history that contains huge trees, for example for git blame.
Does the repository have the same (or very similar) files repeated over and over again at different paths in a single commit? If so, the repository might have a reasonable overall size, but when you check it out it balloons into an enormous working copy. (Taken to an extreme, this is called a "git bomb".)
Does the repository include absurdly long path names? That's probably not going to work well with other tools. One or two hundred characters should be enough, even if you're writing Java.
Are there other bizarre and questionable things in the repository?
Annotated tags pointing at one another in long chains?
Octopus merges with dozens of parents?
Commits with gigantic log messages?