Is there a max trainer level in Magikarp Jump?
Simple question, simple answer: yes.
The maximum Trainer Level is currently 79. It tends to increase whenever there's a major game patch.
A full list of levels, as well as EXP needed to reach them, can be found on Serebii.
Rank Exp Required Max. Magikarp Level
Rank 1 0 Lv. 11
Rank 2 2 Lv. 12
Rank 3 9 Lv. 13
Rank 4 18 Lv. 14
Rank 5 29 Lv. 15
Rank 6 43 Lv. 16
Rank 7 58 Lv. 17
Rank 8 75 Lv. 18
Rank 9 96 Lv. 19
Rank 10 121 Lv. 20
Rank 11 151 Lv. 21
Rank 12 188 Lv. 22
Rank 13 233 Lv. 23
Rank 14 289 Lv. 24
Rank 15 352 Lv. 25
Rank 16 423 Lv. 26
Rank 17 503 Lv. 27
Rank 18 599 Lv. 28
Rank 19 714 Lv. 29
Rank 20 852 Lv. 30
Rank 21 1,022 Lv. 31
Rank 22 1,231 Lv. 32
Rank 23 1,489 Lv. 33
Rank 24 1,780 Lv. 34
Rank 25 2,108 Lv. 35
Rank 26 2,478 Lv. 36
Rank 27 2,922 Lv. 37
Rank 28 3,454 Lv. 38
Rank 29 4,093 Lv. 39
Rank 30 4,879 Lv. 40
Rank 31 5,846 Lv. 41
Rank 32 7,035 Lv. 42
Rank 33 8,376 Lv. 43
Rank 34 9,889 Lv. 44
Rank 35 11,596 Lv. 45
Rank 36 13,644 Lv. 46
Rank 37 16,101 Lv. 47
Rank 38 19,050 Lv. 48
Rank 39 22,677 Lv. 49
Rank 40 27,139 Lv. 50
Rank 41 32,627 Lv. 51
Rank 42 39,487 Lv. 52
Rank 43 48,061 Lv. 53
Rank 44 58,779 Lv. 54
Rank 45 70,869 Lv. 55
Rank 46 84,507 Lv. 56
Rank 47 99,890 Lv. 57
Rank 48 118,350 Lv. 58
Rank 49 140,502 Lv. 59
Rank 50 167,084 Lv. 60
Rank 51 199,780 Lv. 61
Rank 52 239,996 Lv. 62
Rank 53 289,462 Lv. 63
Rank 54 351,294 Lv. 64
Rank 55 428,584 Lv. 65
Rank 56 525,197 Lv. 66
Rank 57 634,176 Lv. 67
Rank 58 757,104 Lv. 68
Rank 59 895,767 Lv. 69
Rank 60 1,062,163 Lv. 70
Rank 61 1,261,838 Lv. 71
Rank 62 1,501,448 Lv. 72
Rank 63 1,796,168 Lv. 73
Rank 64 2,158,674 Lv. 74
Rank 65 2,604,556 Lv. 75
Rank 66 3,161,909 Lv. 76
Rank 67 3,858,600 Lv. 77
Rank 68 4,729,464 Lv. 78
Rank 69 5,711,798 Lv. 79
Rank 70 6,819,871 Lv. 80
Rank 71 8,069,777 Lv. 81
Rank 72 9,569,664 Lv. 82
Rank 73 11,369,529 Lv. 83
Rank 74 13,529,367 Lv. 84
Rank 75 16,185,968 Lv. 85
Rank 76 19,453,587 Lv. 86
Rank 77 23,472,758 Lv. 87
Rank 78 28,496,722 Lv. 88
Rank 79 34,776,677 Lv. 89