jmc

Since the mean of the heights of the 5 tallest buildings in Los Angeles before the new building was built was 733, the sum of their heights must have been $5\cdot 733=3665$. After the new building is built, the shortest of these, which was 625 feet tall, is replaced as a member of the five tallest buildings, which, since it is 885 feet tall, is $885-625=260$ feet taller. Therefore the sum of the heights of the five tallest buildings increases by 260 feet, to $3665+260=3925$ feet. This means that the new mean height of the 5 tallest buildings is $\frac{3925}{5}=785$ feet, so the mean has increased by $785-733=\boxed{52}$ feet. Note that this amount is just the difference in the height of the two buildings divided by 5.