How to calculate column value based on hierarchy

You can use:

d={'Baby':0,'Child':1,'Teenager':2,'Adult':3}
df['rank']=df.Stage.map(d)
df['Highest_Stage_Reached']=df.groupby('Name')['rank'].transform('max').\
                                         map({v: k for k, v in d.items()})
print(df.drop('rank',1))

    Name     Stage Highest_Stage_Reached
0   Adam     Child              Teenager
1  Barry     Child                 Adult
2    Ben     Adult                 Adult
3   Adam  Teenager              Teenager
4  Barry     Adult                 Adult
5    Ben      Baby                 Adult