Does MySQL have a more private and complete catalog other then INFORMATION_SCHEMA?
It funny you asked this. Over 7 years ago (Jul 11, 2011), I answered the post What is the point of the TABLE_CATALOG column in INFORMATION_SCHEMA.TABLES?.

All I mention there is MySQL trying to be SQL-92 compatible, while PostgreSQL does make actual use of the information_schema (it being localized to whatever database you connected, or as I say in the post "The collapse is automatic").

Due to its open storage engine infrastructure, MySQL's exposure of functions allows for installing your own. For example, in the docs you have this:

mysql&gt; CREATE FUNCTION metaphon RETURNS STRING SONAME 'udf_example.so';
mysql&gt; CREATE FUNCTION myfunc_double RETURNS REAL SONAME 'udf_example.so';
mysql&gt; CREATE FUNCTION myfunc_int RETURNS INTEGER SONAME 'udf_example.so';
mysql&gt; CREATE FUNCTION sequence RETURNS INTEGER SONAME 'udf_example.so';
mysql&gt; CREATE FUNCTION lookup RETURNS STRING SONAME 'udf_example.so';
mysql&gt; CREATE FUNCTION reverse_lookup
    -&gt;        RETURNS STRING SONAME 'udf_example.so';
mysql&gt; CREATE AGGREGATE FUNCTION avgcost
    -&gt;        RETURNS REAL SONAME 'udf_example.so';


I have seen statements like these when installing Percona Server, and Percona supplies three commands to run (like these) to expose parts of Percona Toolkit.

Anyway, going back to your actual question: INFORMATION_SCHEMA gives you nothing than what you already know. Everything else is pluggable.