What kind of tree is this — why do its huge leaves change shape and "grow fingers"?
I'm going to take a stab here.  It looks a lot like Artocarpus altilis or breadfruit.  However, the trunk is a bit gnarly for breadfruit, but maybe that is the result of being a street tree.  Also, I can't find any reliable information about the variation in the leaves (heterophylly) for this species.  However, other species in the genus have entire leaves, so perhaps this is hybrid.  Also, Artocarpus chaplasha does show heterophylly related to light intensity and leaf position (online images suggest your tree is not A. chaplasha however).  I think the climate in Taipei is consistent with Artocarpus too.  If it's not breadfruit, I imagine it is a species from the Moraceae.  Finally, in some of your pictures it looks like there is an adjacent tree (possibly a Ficus or Clusia) mixing in -- those leaves are obtuse rather than acute at the tip.
EDIT:
I'm including one of your photos with the leaves of the tree in question circled in yellow, and what I believe to be the leaves of a different tree circled in red (in case my earlier comments were too cryptic).

EDIT 2:
Here is an excerpt of Fig 1 from Jones et al. &quot;Morphological diversity in breadfruit (Artocarpus, Moraceae): insights into domestication, conservation, and cultivar identification&quot; Genetic Resources and Crop Evolution vol 60 no 1 pgs 175-192 (2013) which shows the diversity of leaf morphology in Artocarpus including some hybrids.  I'm convinced you are looking at one of the species in this genus or a hybrid, and looking at the leaves in the panel for A. altilis there is considerable diversity (phenotypic plasticity) w/i that species, so that may be as close as it can be nailed down.
Finally, I can't run it but apparently there is graphical key to Artocarpus species which might be fun to try.

EDIT 3:
@uhoh has verified that all the leaves in the pictures belong to the same tree, so, it's likely we are seeing heterophylly based on the age of the leaves.  I went back to the reference above, and found this sentence would would seem to be consistent with the young leaves being entire and obtuse (and we've already established that the more mature leaves show a lot of variation): &quot;Leaf characteristics, such as the number of lobes, degree of dissection, size, and shape can vary between young shoots and older branches&quot;.  So I think all the observations and literature point in the same direction, namely good 'ol breadfruit.