Far more than seven. There are hundreds. It is a common misconception that the only infallible teachings are solemn definitions made by the Pope (ex cathedra), when in reality the Church teaches infallibly in other ways too. (Klaus’s list of seven above concerns only ex cathedra definitions, and it is a very conservative estimate at that.)
According to Vatican II (LG 25), the Church also teaches infallibly through her bishops agreeing together, not only in ecumenical councils, but also in day-to-day teaching. Note that
not everything in Councils or day-to-day teaching is infallible, but only statements indicating an intention to define formal teaching. This is determined by the language used. To take one example, “If anyone says ____, let him be anathema” (see the canons of the Council of Trent, for example).
In addition to infallible teachings there are also those to which we are bound to give religious submission, even though we do not give the assent of faith (see CIC, Can. 752).
The most comprehensive list I know of, that lists infallible and binding teachings (among other things) is Ludwig Ott’s
Fundamentals of Catholic Dogma. Numerous websites have collected some of these teachings, such as the “Work of God” link above (which unfortunately does not indicate the level of teaching for each statement).
A conservative estimate of dogmas numbers around 255, and I have seen lists that number over 300.
This list (via F. John Loughnan) includes the level of teaching for each statement, but you have to know the Latin abbreviations. “De fide” for example means “of the faith.” That’s the highest level. For more, see Fr. John Trigilio’s
discussion of infallibility.