cómo obtener el conteo de valores en mysql

Si sus datos no son demasiado grandes, puede usar este truco en MySQL:

select user_id, max(dte),
       substring_index(group_concat(place_visited order by cnt desc), ',', 1)
from (select user_id, place_visited, count(*) as cnt, max(recorded_date) as dte
      from user_places
      group by user_id, place_visited
     ) upv
group by user_id;