Not including Chanukah, where we read for 8 days, or Pesach and Sukkos, where it is read for eight and nine days respectively (in Israel, 7 and 8).
If Rosh Hashanah falls on Thursday, then both days of Rosh Hashanah the Torah reading for R.H. would be read, the following day would be Shabbos so the Parsha would be read, and Tzom Gedaliah would be pushed to Sunday, so that would be the fourth day. Additionally, the next morning on Monday the Torah would also be read, so that's 5 days in a row!
If Rosh Hashanah falls out Tuesday and Wednesday or Monday and Tuesday, then it will be exactly 4 days consecutively (because of 2 days R.H., Tzom Gedaliah, and Thursday or Monday).
If Rosh Chodesh is Tuesday and Wednesday, that would be another case of exactly 4 days because of the Monday and Thursday readings.