Use loops!
odd_nums = []
for i in xrange(10):
value = int(raw_input('Enter an integer: '))
if value % 2 != 0:
odd_nums.append(value)
if len(odd_nums) != 0:
print max(odd_nums)
else:
print "No odd values"
How to read this line:
for i in xrange(10):
for i in xrange(10):
This line means starting at 0 (the default starting point for xrange()), and ending at 10-1 (upper bound is not included), execute the following code with i set to the current iteration of the loop. This executes the code exactly 10 times (i = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9).
The rest was made as similar to your code as possible, and so should be understandable.
Edit: changed input to value, as per comments, as well as added some explanation.