What is the maximum number of overs with the same ball in a cricket test match. I have asked this question on google and not given satisfactory answer, could you please advise of law.
There is no hard limit. So long as the ball is fit for play, any number of overs (in one innings; the ball is always replaced at the start of every innings1) may be bowled with it should the fielding captain wish to do so. However, should a ball which is more than 110 overs old become unfit for play, it must be replaced with a new ball; quoting from Law 4.5 of the current Test Match playing conditions:
If, during play, the ball cannot be found or recovered or the umpires agree that it has become unfit for play through normal use, the umpires shall replace it with a ball which has had wear comparable with that which the previous ball had received before the need for its replacement. When the ball is replaced, the umpires shall inform the batsmen and the fielding captain.
However, if the ball needs to be replaced after 110 overs for any of the reasons above, it shall be replaced by a new ball. If the ball is to be replaced, the umpires shall inform the batsmen, the fielding captain and the scorers.
The second paragraph here is an addition to the general Law 4. This change to the playing conditions occurred after a couple of incidents where teams did want to carry on bowling with an very old ball; once the old ball went out of shape, it practically became almost impossible to find a ball "which has had wear comparable with that which the previous ball had received".
1. This is a theoretical difference between the general Laws and the Test Match playing conditions; see this answer for more details.