In order to expand the popular tags list (shown to the right), you have to edit the mod_tags_popular.xml file.
It can be found in the root directory of Joomla, under modules/mod_tags_popular/mod_tags_popular.xml.
Access the directory with your (web) file manager (or via FTP), make a copy of the file (just in case), and around line 30 you should find the following code:
<field
name="maximum"
type="integer"
default="5"
first="1"
last="20"
step="1"
label="MOD_TAGS_POPULAR_MAX_LABEL"
description="MOD_TAGS_POPULAR_MAX_DESC">
</field>
type="integer"
default="5"
first="1"
last="20"
step="1"
label="MOD_TAGS_POPULAR_MAX_LABEL"
description="MOD_TAGS_POPULAR_MAX_DESC">
</field>
Modify the value under "last" (as shown in bold above or marked yellow below):
