Is there a conjecture with maximal prime gaps
I haven't seen a conjecture of this type. It is conjectured that among all primes up to $x$, the largest gap has size like $(\ln x)^2$ or a constant multiple thereof. The next time that gap occurs, each number following the gap will have a roughly $1/\ln x$ probability of being prime (by the prime number theorem). So we expect the next gap to be about $Y\ln x$ larger than the previous one, where $Y$ is a continuous random variable with a Poisson distribution with parameter $\lambda=1$. This implies that the order of magnitude of $M_{N+1}-M_n$ typically has size $\sqrt M_n$ (times a fluctuating constant); in particular, for any $\varepsilon>0$, we should have $M_{n+1}/M_n < 1+\varepsilon$ for sufficiently large $n$.
I pulled one of the tables of prime gaps off wikipedia and put a final column, $g/\log^2p,$ just as in the section in Guy's book. For $11 \leq p < 4 \cdot 10^{18},$ we have $g < \log^2p.$ Completely unprovable for larger $p.$ After line 3 (prime is 7) the closest we get to $1$ is line 64, $ \; g = 1132,$ $ \; p \approx 1.69 \cdot 10^{15},$ $ \; g/\log^2p \approx 0.920639.$ I believe Cramer-Granville is the conjecture that $ \; \limsup g/\log^2 p$ is nonzero but finite, and the disagreement is over whether it is more likely to be $1$ or something else.
Stolen from
http://users.cybercity.dk/~dsl522332/math/primegaps/maximal.htm
the size of the gap is g
next are the number of decimal digits in p
for 4 * 10^18 > p >= 11, g < log^2 p = (log p)^2.
Oh, logarithms base e == 2.718281828459
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g digits of p p log p g/log p g/log^2 p
1 1 1 2 0.693147 1.4427 2.08137
2 2 1 3 1.09861 1.82048 1.65707
3 4 1 7 1.94591 2.05559 1.05637
4 6 2 23 3.13549 1.91357 0.610294
5 8 2 89 4.48864 1.78228 0.397065
6 14 3 113 4.72739 2.96147 0.626449
7 18 3 523 6.25958 2.87559 0.45939
8 20 3 887 6.78784 2.94644 0.434076
9 22 4 1129 7.02909 3.12985 0.445271
10 34 4 1327 7.19068 4.72835 0.657566
11 36 4 9551 9.1644 3.92824 0.428642
12 44 5 15683 9.66033 4.55471 0.471486
13 52 5 19609 9.88374 5.26116 0.532305
14 72 5 31397 10.3545 6.95352 0.671548
15 86 6 155921 11.9571 7.19238 0.601515
16 96 6 360653 12.7957 7.50254 0.586334
17 112 6 370261 12.822 8.73501 0.681254
18 114 6 492113 13.1065 8.698 0.663642
19 118 7 1349533 14.1153 8.35974 0.592248
20 132 7 1357201 14.1209 9.34782 0.661983
21 148 7 2010733 14.514 10.197 0.702566
22 154 7 4652353 15.3529 10.0307 0.653342
23 180 8 17051707 16.6518 10.8097 0.649161
24 210 8 20831323 16.852 12.4615 0.739466
25 220 8 47326693 17.6726 12.4487 0.704405
26 222 9 122164747 18.6209 11.9221 0.640254
27 234 9 189695659 19.0609 12.2764 0.644062
28 248 9 191912783 19.0726 13.003 0.681764
29 250 9 387096133 19.7742 12.6427 0.639356
30 282 9 436273009 19.8938 14.1753 0.712549
31 288 10 1294268491 20.9812 13.7266 0.654231
32 292 10 1453168141 21.097 13.8408 0.656056
33 320 10 2300942549 21.5566 14.8447 0.688637
34 336 10 3842610773 22.0694 15.2247 0.689855
35 354 10 4302407359 22.1824 15.9586 0.719423
36 382 11 10726904659 23.096 16.5396 0.716125
37 384 11 20678048297 23.7523 16.1668 0.680642
38 394 11 22367084959 23.8309 16.5332 0.693772
39 456 11 25056082087 23.9444 19.0441 0.795349
40 464 11 42652618343 24.4764 18.9571 0.774506
41 468 12 127976334671 25.5751 18.299 0.715502
42 474 12 182226896239 25.9285 18.281 0.705055
43 486 12 241160624143 26.2087 18.5434 0.707529
44 490 12 297501075799 26.4187 18.5475 0.702059
45 500 12 303371455241 26.4382 18.912 0.715328
46 514 12 304599508537 26.4423 19.4386 0.735133
47 516 12 416608695821 26.7554 19.2858 0.720819
48 532 12 461690510011 26.8582 19.8078 0.737495
49 534 12 614487453523 27.1441 19.6728 0.724756
50 540 12 738832927927 27.3283 19.7597 0.723048
51 582 13 1346294310749 27.9284 20.839 0.746159
52 588 13 1408695493609 27.9737 21.0198 0.751412
53 602 13 1968188556461 28.3081 21.266 0.751232
54 652 13 2614941710599 28.5923 22.8034 0.797536
55 674 13 7177162611713 29.6019 22.7688 0.769166
56 716 14 13829048559701 30.2578 23.6633 0.782057
57 766 14 19581334192423 30.6056 25.0281 0.817762
58 778 14 42842283925351 31.3885 24.7861 0.789655
59 804 14 90874329411493 32.1405 25.0152 0.778307
60 806 15 171231342420521 32.774 24.5926 0.750369
61 906 15 218209405436543 33.0165 27.4408 0.831126
62 916 16 1189459969825483 34.7123 26.3884 0.760203
63 924 16 1686994940955803 35.0617 26.3535 0.751632
64 1132 16 1693182318746371 35.0654 32.2825 0.920639
65 1184 17 43841547845541059 38.3194 30.8982 0.806335
66 1198 17 55350776431903243 38.5525 31.0745 0.806032
67 1220 17 80873624627234849 38.9317 31.337 0.804922
68 1224 18 203986478517455989 39.8568 30.7099 0.770506
69 1248 18 218034721194214273 39.9234 31.2598 0.782995
70 1272 18 305405826521087869 40.2604 31.5943 0.784749
71 1328 18 352521223451364323 40.4039 32.8681 0.813489
72 1356 18 401429925999153707 40.5338 33.4536 0.825325
73 1370 18 418032645936712127 40.5743 33.7652 0.832181
74 1442 18 804212830686677669 41.2286 34.9757 0.848335
75 1476 19 1425172824437699411 41.8008 35.3103 0.844728
g digits of p p log p g/log p g/log^2 p
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Here's a heuristic analysis using Cramér's model. TL;DR: the conjecture is likely false.
The expected maximal gap in this model is $\log^2n$, so suppose we're looking just after finding just such a gap. The probability that a number is prime is $x=1/\log n$ and so the probability that a given prime will have a gap of length $k+1$ is $x(1-x)^k$. The probability that a given prime will not be followed by a record gap is thus $$ 1-(1-x)^{\log^2n}=1-\left((1-1/\log n)^{\log n}\right)^{\log n}\approx1-e^{-\log n}=1-1/n. $$ and the probability that a given prime will be followed by a record of at least $k$ times the old one is $$ (1-x)^{k\log^2n}\approx e^{-k\log n}=n^{-k}. $$
Combining the two, the probability that the gap will be exceeded by a factor of $k$ or more is $$ \sum_{i=0}^\infty(1-1/n)^in^{-k}=\frac{n}{n^k}=\frac{1}{n^{k-1}} $$ and hence $k=2$ is right on the boundary: we expect records to double the previous record infinitely often (since $\int1/n$ diverges), but records should be 2.001 times the previous record finitely often.
Now this is really taking the heuristic too far. Cramér's model is not state of the art, and has been shown to make incorrect predictions on the fine behavior of the primes ([1], [2]). But the basic idea is that there is a phase transition, probably around 2, from ratios that appear infinitely often to finitely often (or never!).
Improvement
A better version would allow the starting gap to be other than $\log^2n$. This crude version is conservative, since a distribution of values would make it easier to get large factors between the two.
In fact, if you redo the calculation supposing that the old record is only $s\log^2 n$ then you expect that gaps of size $1+1/s$ should occur infinitely often. So if you think that a positive proportion of the time the largest prime gap below $n$ is $0.99\log^2n$ then you should get records topping the previous one by a factor of 2+1/99 infinitely often.
Technical Notes
A further improvement would be to take small factors into account (a relative of the so-called W-trick). It's hard to predict the net effect but if anything it would also make larger factors happen more often.
A minor issue is that my analysis uses $1/\log n$ as though it is constant. But the interesting range is the next $n$ primes after $n$, so it suffices to look up to about $n+n\log n$ which has logarithm about $\log n+\log\log n$ which is less than $(1+\varepsilon)\log n$ for any $\varepsilon>0$ and large enough $n$.
Conclusion
The conjecture is on shaky ground. There's good reason to think big ratios of maximal gaps happen infinitely often. On the other hand, looking at the heuristics we're talking about amazingly big numbers before these sorts of events happen. Each event we're looking at is a new maximal prime gap, and a 'success' at any given maximal prime gap has asymptotic probability 0 of happening, getting positive probability only when we integrate over a large number of new maximal prime gaps. But we know only a handful of these, so it's entirely possible that we'll never see only of these megajumps.
Bibliography
[1] János Pintz, Cramér vs. Cramér. On Cramér's probabilistic model for primes, Funct. Approx. Comment. Math. 37 (2007), part 2, pp. 361–376.
[2] H. Maier, Primes in short intervals, Michigan Math. J. 32 (1985), pp. 221–225.
This is all some rather shaky heuristics, but here is what I think:
A random integer n is prime with probability $\frac{1}{\ln{n}}$ and composite with probability $1 - \frac{1}{\ln{n}}$.
A random integer n is followed by at least $M$ composite numbers with probability $(1 - \frac{1}{\ln{n}})^M$ which is about $\exp (-\frac{M}{\ln{n}})$. That's true for every integer $n$, including integers $n$ which are primes.
If the record gap so far is $M$, then a prime $p$ is followed by a record gap with probability $\exp (-\frac{M}{\ln{p}})$. It is followed by a gap of length $2M$ or more with probability $\exp (-\frac{2M}{\ln{p}})$. If that prime $p$ is indeed followed by a record gap, then the probability that it is followed by a gap doubling the record is $\exp (-\frac{M}{\ln{p}})$.
So the chance that a prime $p$ is followed by a record gap is the same as the chance that the record gap is twice the previous record gap. The table above shows 75 record gap up to $1.4 * 10^{18}$. The distance between two record gaps is about $8 * 10^{17}$. $\ln p$ is about 40, so the chance for a prime being followed by a record gap is about $\frac{1}{2 * 10^{16}}$. The chance that a record gap is twice the previous gap is also $\frac{1}{2 * 10^{16}}$. So this is very unlikely to happen.
Anyone who knows how to make $\frac{1}{\ln{p}}$ look nicer? See comment below.