While it is possible for a wooden building to exceed 200 feet (approx. 61 metres) in height (see, for example, Pagoda of Fogong Temple from 1056) there is nothing approaching this height from medieval Europe still standing. 5 or 6 floors seems to have been the limit, with two to four-storey buildings common depending on the city and the century.
Highest / Biggest
Many of the tallest medieval half-timbered houses still standing are found in Germany and France. In some cases, though, only the foundations or ground floor date from medieval times - these have been excluded from the images below.
Webergasse 8b, Esslingen am Neckar (1266-67). According to this article, "the entire core construction is still preserved, that is, the beams made of oak wood are around 750 years old." Image source.
Nos. 11 & 13 rue Francois Miron are believed to be the oldest houses in Paris. "Sources indicate that they could have been built as early as the 14th century." The frame type suggests they were built before the 1450s as this style had gone out of use by then. Image source.
Hafenmarkt 4-10, Esslingen am Neckar. Dendrochronology shows that this building dates from between 1328 and 1331. Image source.
Based on the evidence of structures built before 1453 which are still standing, 5 floors (6 if one includes attic rooms) was the limit for half-timber buildings. If we assume an average of around 10 feet per floor (including construction between floors), this gives us a total height of up to 50 feet, though some upper floors were usually less.
Finally, the Maison d'Adam in Angers (France), built shortly after 1491, falls a little outside your time-frame but it is 15th century and impressive enough to merit a mention (and a picture):
Average number of floors
This is difficult to answer as it depended on where, when and how wealthy. However, the chronicler Matthew Paris notes that, on a visit to Paris in 1254, King Henry III of England
was much struck with `the elegance of the houses which were made of
plaster, and were threechambered and even of four or more stories
Cited in L. F. Salzman, Building in England Down to 1540
the king 'was struck' because in London at that time, town houses were usually only two storeys. However,
by the late thirteenth century there were houses in Cheapside of three
storeys plus a garret.
Source: D. M. Palliser (ed), The Cambridge Urban History of Britain: volume I, 600-1540
Also, in London, in
About 1300 it was thought that houses in the city commonly had two or
three storeys over a cellar and were divided into several units of
Source: D. M. Palliser (ed)