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A small bump on the anus that is not painful but getting larger over the course of a day would most often be a hemorrhoid. A hemorrhoid can be inflamed resulting in pain or other symptoms, but there are many people in whom a hemorrhoid does not cause any symptoms other than the bump.
It is also possible to get a cyst or an abscess, but a cyst or abscess that doubles in size in a day will typically have pain or other symptoms.
For hemorrhoids that are not causing symptoms, the usual intervention are more oriented towards having bowel movements that will pass easily, including increased fiber intake and possibly a stool softener, such as Colace. The over the counter hemorrhoid creams are more oriented towards relief of inflammation or pain associated with hemorrhoid, and while these are fine to use, they usually are not as helpful as interventions to assure that the stools will pass easily. It also may promote healing to apply moist heat, although it is usually easiest to do this by soaking in hot bath water. Soaking may also help with cysts, but if there is a worsening cyst or an abscess, it would usually require drainage by a doctor and possibly antibiotics.
If I can provide any additional information, please let me know.
It is impossible to predict. Many hemorrhoids resolve within a few days, but there are other hemorrhoids that can become a chronic problem. It is true that the likelihood of chronic hemorrhoids is particularly greater in people that have chronic problems with difficult bowel movements, but it is not always true.