Solving Code Jam's Dice Straight
Your code looks good, seems to work and is somehow documented but it can still be improved in various ways (not only performances which will be only tackled at the end if at all).

Code organisation

Your call to main could be moved behind an if __name__ == "__main__": guard in case you ever want to reuse get_straight(). It may not be relevant in your case but it is a good habit to take.

Then, it may be a good idea to split your code into smaller functions, easier to understand and to maintain.

By doing so, you'd also separate the concerns my having the code computing the results you need not mixed up with the logic handling the input/output.

Finally, and maybe even more important, that will make your code testable. Indeed, before performing any changes in the algorithm, it is highly recommended to write unit-tests to prevent you from breaking everything (tests are usually not fail-proof but they help a lot).

from operator import itemgetter
from itertools import groupby    


def get_straight(chunk: list, idx: int, straight: int, dice: set) -&gt; int:
    """Get max straight starting at index idx."""
    if idx == len(chunk) - 1:
        return straight

    item = chunk[idx]

    return max((get_straight(chunk, idx+1, straight+1, dice-{x})
                for x in item[1] if x in dice), default=straight)


def longest_straight_length(dices):
    """Return the length of the longest possible straight using `dices`."""
    integers = []
    for j, dice in enumerate(dices):
        integers.extend((val, j) for val in dice)
    integers.sort(key=itemgetter(0))

    grouped = [(k, [x[1] for x in g])
               for k, g in groupby(integers, itemgetter(0))]
    chunks = []

    # find runs of consecutive numbers
    for k, g in groupby(enumerate(grouped), lambda x: x[0]-x[1][0]):
        g = list(g)
        if len(g) &gt; 1:
            chunks.append(list(map(itemgetter(1), g)))

    chunks.sort(key=len, reverse=True)

    max_straight = 1

    for chunk in chunks:
        if len(chunk) &lt;= max_straight:
            break

        for k in range(len(chunk)-1):
            if len(chunk) - k &lt; max_straight:
                break
            dice = set(range(len(dices)))
            straight = get_straight(chunk, k, 0, dice)
            if straight &gt; max_straight:
                max_straight = straight
    return max_straight

def interactive():
    for i in range(int(input())):
        ndice = int(input())
        dices = [[int(x) for x in input().split()] for _ in range(ndice)]
        print("Case #{}: {}".format(i+1, longest_straight_length(dices)))


def unit_tests():
    test_cases = [
        ([[4, 8, 15, 16, 23, 42],
          [8, 6, 7, 5, 30, 9],
          [1, 2, 3, 4, 55, 6],
          [2, 10, 18, 36, 54, 86]],
         4),
        ([[1, 2, 3, 4, 5, 6],
          [60, 50, 40, 30, 20, 10]],
         1),
        ([[1, 2, 3, 4, 5, 6],
          [1, 2, 3, 4, 5, 6],
          [1, 4, 2, 6, 5, 3]],
         3)
    ]
    for dices, expected_result in test_cases:
        actual_result = longest_straight_length(dices)
        if actual_result != expected_result:
            print(expected_result, actual_result, dices)

if __name__ == "__main__":
    if True:
        unit_tests()
    else:
        interactive()


Rewriting the logic

The beginning of the logic creating chunks could be rewritten using a dictionnary rather than performing smart groupby and tricky index accesses. (At the end, I re-build the data structure you are using to be able to keep the code you've written). We'd get something like:

def longest_straight_length(dices):
    """Return the length of the longest possible straight using `dices`."""
    # Mapping value to list of dice indices
    number_on_dices = dict()
    for i, dice in enumerate(dices):
        for val in dice:
            number_on_dices.setdefault(val, []).append(i)

    # Consecutive chunks
    chunks = []
    for k, g in groupby(enumerate(sorted(number_on_dices.keys())), lambda x: x[0] - x[1]):
        g = list(g)
        if len(g) &gt; 1:
            chunk = [y for (_, y) in g]
            chunks.append(chunk)
    chunks.sort(key=len, reverse=True)

    # Re-add values to be reuse existing logic for the time being
    chunks = [[(val, number_on_dices[val]) for val in chunk] for chunk in chunks]

    max_straight = 1

    for chunk in chunks:
        if len(chunk) &lt;= max_straight:
            break

        for k in range(len(chunk)-1):
            if len(chunk) - k &lt; max_straight:
                break
            dice = set(range(len(dices)))
            straight = get_straight(chunk, k, 0, dice)
            if straight &gt; max_straight:
                max_straight = straight
    return max_straight


Then, it appears clearer that the first element in pairs from chunks are never used.
We can simply have:

chunks = [[number_on_dices[val] for val in chunk] for chunk in chunks]


and in get_straight:

return max((get_straight(chunk, idx+1, straight+1, dice-{x})
            for x in item if x in dice), default=straight)


Ultimately, we can simplify the code building chunks:

# Consecutive chunks
chunks = []
for k, g in groupby(enumerate(sorted(number_on_dices.keys())), lambda x: x[0] - x[1]):
    g = list(g)
    if len(g) &gt; 1:
        chunk = [number_on_dices[y] for (_, y) in g]
        chunks.append(chunk)
chunks.sort(key=len, reverse=True)


However, the expensive part is still here, in the recursive calls to get_straight. (I haven't found a trick for this yet but current version of the code is available here: http://termbin.com/yg5l ).

Bug ?

As I was trying to come up with bigger and bigger examples with more recursive calls, I've added the following test cases:

    ([[1, 2, 3, 4, 5, 6],
      [7, 2, 3, 4, 5, 6],
      [7, 8, 3, 4, 5, 6],
      [7, 8, 9, 4, 5, 6],
      [7, 8, 9,10, 5, 6],
      [7, 8, 9,10,11, 6],
      [7, 8, 9,10,11,12]],
     7),
    ([[1, 1, 1, 1, 1, 1],
      [2, 2, 2, 2, 2, 2],
      [3, 3, 3, 3, 3, 3],
      [4, 4, 4, 4, 4, 4],
      [5, 5, 5, 5, 5, 5],
      [6, 6, 6, 6, 6, 6],
      [7, 7, 7, 7, 7, 7]],
     7),


In both cases, we could generate the sequence [1, 2, 3, 4, 5, 6, 7] of length 7. However, the code returned 6. If you want to go further from that part, the code with additional tests and performance indicators is available here: http://termbin.com/c690 .