The new update of this article from Joe O'Brien - he showed that there is a hand that will give the pone a total of 24 in pegging.
The Pone holding 5 5 4 4, and the Dealer holding 5 4 4 (na/). The Pone hits both the fifteen and the thirty-one, picking up a triple at the 15 mark and a pair and four of a kind on the way to the thirty-one mark. This all adds up to 24.
Play goes like this:

Pone Dealer Pone Points Dealer Points Card Count
15 25   2 10
35 44 8 - Fifteen for 2 + 6 for tripple (8)   19
54 64 2 - For the pair (10) 6 27
74 n/a 14 - For 4 of a kind + two for 31 (24)   31
Red - succession of play

Total Pone pegging: 8 + 2 + 14 = 24 pts.

Below is a previous version of this article about maximum Pone pegging score

This article has been updated with the help of our player Kenneth Anderson - he showed, that the maximum cribbage scoring for Pone pegging is 23. Here is how.
Pone's hand: 5 (or 6) 4 4 4
Dealer's hand: 5 (or 6) 4 9 9 (9 9 could be any cards worth 9 or above)
Play goes like this:

Pone Dealer Pone Points Dealer Points
6 5 0 0
4 4 pegs 5: 3 for run of 3, 2 for the 15 pegs 2 for the pair
4 GO pegs 6 for the pair royal 0
4 (NA) pegs 12 for the double pair royal  
GO (NA)    
Total Pone points: 5+6+12 = 23
Pone pegs 23 in this scenario.

Below is our original article about maximum Pone pegging score

Many sites and cribbage articles state, that pone maximal pegging is 18. A cribbage player Kyle Graham showed, that it is incorrect and Pone can score more - 22 points. Here are examples of cribbage Pone hands:

Pone’s hand: A A 3 3
Dealer’s hand: A 3 3 (NA) – the last card does not matter

Pone Dealer Pone Points Dealer Points
A A 0 2
A 3 6 0
3 3 2 6
3 (NA) 14 (12 pts for 4 the in the sequence of pairs and 2 pts for making 15) 6
Total Pone pegging: 6+2+14 = 22 pts

If you were to substitute the 3s in each player’s hand, a 4, 5, or 6 would make 20 pts. Using a 7 would make 22 points as well because the pone would score 2 pts for making 31 instead of 15. In addition, you could substitute other cards for the A as well, so long as all cards, with the exception of the dealer’s last card equal 31 or less. In any case, using these sequences, pone scores at least 20 points, and 22 points when making either 15 or 31.

Another way to score 22 points:
Pone’s hand: A A 7 7
Dealer’s hand: A 7 7 (NA) – the last card does not matter

Pone Dealer Pone Points Dealer Points
A A 0 2
A 7 6 0
7 7 2 6
7 (NA) 14 (12 pts for 4 the in the sequence of pairs and 2 pts for making 31) 6
Total Pone pegging: 6+2+14 = 22 pts

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