Multiple inline functions in Oracle

Yes it is possible you should replace SELECT func() INTO variable with direct function call:

ORA-06553: PLS-231: function 'ADD_STRING' may not be used in SQL

with 
  function add_string(p_string in varchar2) return varchar2
  is
    --Function to add a string
    l_buffer varchar2(32767);
  begin
    l_buffer := p_string || ' works!';
    --
    return l_buffer;
    --
  end ;

  function doesnt_it(p_string in varchar2) return varchar2
  is 
    l_buffer varchar2(32767);
    -- pp_string varchar2(32767);
  begin
    -- select add_string(p_string) into pp_string from dual;
    l_buffer := add_string(p_string) || ' Doesnt it?';  -- here is function call
    --
    return l_buffer;
    --
  end ; 

select  doesnt_it('Yes, it') as outVal
from dual;

db<>fiddle demo

Output:

OUTVAL
Yes, it works! Doesnt it?

Alternative solution:

pp_string :=  add_string(p_string);
l_buffer := pp_string || ' Doesnt it?';

db<>fiddle demo2

You could also have inline function/procedure in inline with block Procedures in the WITH Clause