ValueError: cannot insert ID, already exists

Use parameter drop=True which not create new column with index but remove it:

result = df.groupby(['ID']).agg({'TIME': 'mean', 'ID': 'count'}).reset_index(drop=True)
print (result)
   ID      TIME
0   3  2.666667
1   1  3.000000

But if need new column from index need rename old column names first:

result = df.groupby(['ID']).agg({'TIME': 'mean', 'ID': 'count'})
           .rename(columns={'ID':'COUNT','TIME':'MEAN_TIME'})
           .reset_index()
print (result)
   ID  COUNT  MEAN_TIME
0   1      3   2.666667
1   2      1   3.000000

Solution if need aggreagate by multiple columns:

result = df.groupby(['ID']).agg({'TIME':{'MEAN_TIME': 'mean'}, 'ID': {'COUNT': 'count'}})
result.columns = result.columns.droplevel(0)
print (result.reset_index())
   ID  COUNT  MEAN_TIME
0   1      3   2.666667
1   2      1   3.000000