Does this game terminate?
Regarding Beggar-My-Neighbour

Paulhus (1, p.164) wrote in 1999:


  If $C$ is a full deck of cards, does ${D_{2}}^{'}(C)$ have a cycle? We
  leave  this question unanswered except to say that we have been unable
  to find one in 
  3.2 billion randomly chosen deals.


But Conway et al. (2, p.892) wrote in 2006:


  Strip-Jack-Naked, or Beggar-My-Neighbour **1
  
  Another problem that took almost 47 years to solve concerns this old 
  children’s game. Each of the two players starts with about half of the
  cards  (held face-down), which they alternately turn over onto a
  face-upwards “stack”  on the table, until one of them (who's now “the
  commander”) first deals one of  the “commanding cards” (Jack, Queen,
  King, or Ace).
  
  After one of these has been dealt, the other player (now “the
  responder”) turns  over cards continuously until EITHER. **2 a new
  commanding card appears (when  the players change roles **3) or
  respectively 1, 2, 3, or 4 non-commanding cards  have been turned
  over. In the latter case, the commander turns over the stack  and
  ajoins it to the bottom of his hand. The responder then starts the
  formation  of a new stack by turning over his next card, and play
  continues as before.
  
  A player who acquires all the cards is the winner and in real games,
  it seems  that someone always does win. The interesting mathematical
  question, posed by  one of us many years ago, was “is it really true
  that the game always ends?”  Marc Paulhus has recently found the
  answer to be “no!”. About 1 in 150,000 games  (played with the usual
  52 cards) goes on forever.
  
  We are fairly confident that no one person has played the game
  anything like  that number of times, so the chance (with random
  shuffling) of experiencing a  non-terminating game in a lifetime’s
  play must be very small indeed.
  
  Just as surely, however, the total number of times this game has been
  played by  the World’s **4 children must be significantly larger than
  150,000, so many of  them will have been theoretically non-terminating
  ones. We imagine, though, that  in practice most of them actually did
  terminate because someone made a mistake.


Unfortunately I was not able to found in (2) any reference to the discovery of Paulhus... I would love to see a sequence of cards that gives a non-terminating game in order to say that the problem is solved.

In 2013, Lakshtanov and Aleksenko (3) wrote:


  For card games of the Beggar-My-Neighbor type, we prove finiteness of
  the mathematical expectation of the game duration under the conditions
  that a player to play the first card is chosen randomly and that cards
  in a pile are shuffled before being placed to the deck. The result is
  also valid for general-type modifications of the game rules. In other
  words, we show that the graph of the Markov chain for the
  Beggar-My-Neighbor game is absorbing; i.e., from any vertex there is
  at least one path leading to the end of the game.


but their rules are not the ones I followed when I played the game when I was a child ;-)

To the best of my knowledge the longest Beggar-my-Neighbour game was found in 2014 by William Rucklidge with 7960 cards:

1: -J------Q------AAA-----QQ-
2: K----JA-----------KQ-K-JJK


Regarding Cavacamicia 

I usually played it with a 40 cards deck, simulations with an half deck (only 20 cards) gives 16 non terminating games on a total of 3.448.400 games.

Bibliography

(1)
PAULHUS, Marc M. Beggar my neighbour. American Mathematical Monthly, 1999, 162-165.
http://www.jstor.org/stable/2589054

(2)
BERLEKAMP, Elwyn R.; CONWAY, John H.; GUY, Richard K. Winning Ways for Your Mathematical Plays, Volume 4. AMC, 2003, 10: 12.
http://www.maa.org/publications/maa-reviews/winning-ways-for-your-mathematical-plays-volume-4

(3)
LAKSHTANOV, Evgenii Leonidovich; ALEKSENKO, Alena Il'inichna. Finiteness in the Beggar-My-Neighbor card game. Problems of Information Transmission, 2013, 49.2: 163-166.
http://dx.doi.org/10.1134/S0032946013020051