Try

select *, (select count(*) + 1 from tbl t where t.time < m.time) as Rank from tbl m where name = 'xxxx'

or

select m.name, count(*) + 1 as Rank from tbl m join tbl t on t.time < m.time where m.name...

That query with the where clause will return one line.  However, also note that he requests

People should be able to submit their times and they will show up in a...

select *, (select count(*) from tbl t where t.time <= m.time) as Rank from tbl m where name = 'xxxx'

or

select m.name, count(*) as Rank from tbl m join tbl t on t.time <= m.time where m.name = 'xxxx' group...