You are looking for the table's AUTO_INCREMENT value. MySQL stores such metadata in the INFORMATION_SCHEMA database. The AUTO_INCREMENT values can be found in the TABLES table. The table contains one entry for each database and table.
I don't think Rails or the MySQL gem provide any built-in method for fetching it.
I have used something like this in one of my previous projects:
# config/initializers/auto_increment.rb
module AutoIncrement
  def auto_increment_value
    connection.execute(<<-SQL.squish).first[0]
      SELECT `AUTO_INCREMENT`
        FROM `INFORMATION_SCHEMA`.`TABLES`
       WHERE `TABLE_SCHEMA` = '#{connection.current_database}'
         AND `TABLE_NAME` = '#{table_name}'
    SQL
  end
end
ActiveRecord::Base.extend(AutoIncrement)
You can then execute:
MyModel.auto_increment_value
and it will return the current value for the table's AUTO_INCREMENT value, e.g. 11.
Note that it is not safe to use that value as an explicit ID for your record. You should let MySQL handle the assignment of new IDs – that's what AUTO_INCREMENT is for.