MariaDB does not have ANY_VALUE() function.

For the error, please run CHECK TABLE mysql.proc, most likely it is actually corrupted. Since MariaDB does not have the internal function, it attempts to read mysql.proc to see if there is one in there, and hits the error. When you remove ANY_VALUE, the query does not make server read from mysql.proc, hence no error.

I ran check table mysql.proc and it tests ok.

I wonder if there is some way to get around this problem. I know nothing about managing mysql functions but I was wondering whether it is possible to define some code that checks whether this function is defined and - if not - defines it as a function that just returns its input.

For mysql.proc problem, try also CHECK TABLE mysql.proc EXTENDED. If it doesn't reveal anything, please upload mysql/proc.frm, mysql/proc.MYD and mysql/proc.MYI to ftp.askmonty.org/private.

For the question about a function, you'll obviously need to learn something about managing MySQL functions in order to do that. Documentation is a good start.
MariaDB KB: https://mariadb.com/kb/en/mariadb/user-defined-functions/
MySQL manual: http://dev.mysql.com/doc/refman/5.6/en/adding-functions.html
You can also discuss the question on a mailing list.

CHECK TABLE mysql.proc EXTENDED doesn't give an error either.

I tried to upload the files but it is denied. I tried other directories on this server too - but with the same result.

I will have a look at the links.

Actually, files got uploaded just fine, maybe you tried to do it twice, then the second attempt would fail.
You are getting the error because apparently you didn't run mysql_upgrade after upgrading the server from whatever version you had before. Running mysql_upgrade fixes this problem (and is highly recommended, because there might be other problems).

I'll convert this bug report to a feature request to consider porting ANY_VALUE function from MySQL 5.7 to new versions of MariaDB. This is not a guarantee it will actually happen.

Any update on this? Makes having to code for 2 separate database back-ends for basic querying.

MariaDB 10.3 brings custom aggregate functions - so you can (sort of) do this for yourself:

DROP FUNCTION IF EXISTS ANY_VALUE;

DELIMITER //

CREATE AGGREGATE FUNCTION ANY_VALUE (x LONGBLOB) RETURNS LONGBLOB

BEGIN

 LOOP

  FETCH GROUP NEXT ROW;

  RETURN x;

 END LOOP; 

END //

DELIMITER ;

...assuming that any type can be safely cast to longblob and back again (right? & yes, "ewww....") then I think this should do what you need! Be warned that this always acts like an aggregate function so using it in queries without a GROUP BY will not give the same result as MySQL 5.7.