Some experts have a hunch that for any nonnegative integer $r$ there are infinitely many elliptic curves over $\Bbb Q$ with Mordell-Weil rank at least $r$.
The best empirical evidence for this hunch can be found in Andrej Dujella's tables here and here, the strongest of this evidence being provided by Elkies using constructions involving K3 surfaces.
The only heuristic evidence I know of supporting this hunch is the results of Tate and Shafarevich (strengthened by Ulmer) that the Mordell-Weil ranks of elliptic curves over the a fixed function field $\Bbb F_q(t)$ can be arbitrarily large.
What other heuristic evidence (if any) is there that the ranks of elliptic curves over $\Bbb Q$ should be unbounded?
Seeing as how some experts do no believe this conjecture, I'd also accept answer to the companion question:
What heuristic evidence (if any) is there that the ranks of elliptic curves over $\Bbb Q$ should be uniformly bounded?
Best Answer
Remke remarks: "I guess that the main evidence is the fact that every now and then someone comes up with a new record."
As a not-entirely-serious quantification of that remark, I graphed the following data: For each year starting in 1974 in which a new E(Q) rank record was found, graph the point (year,highest rank found that year). Here's the data: (1974,6),(1975,7),(1977,9),(1982,12),(1986,14),(1992,19), (1993,20),(1994,21),(1997,22),(1998,23),(2000,24),(2006,28),which I took from http://web.math.hr/~duje/tors/rankhist.html. The data looks remarkably linear. The linear correlation coefficient is 0.998. The slope is 0.686. Some of the linearity probably comes from Moore's law (log-linear increase in computing power), but the table also represents major algorithmic advances of Mestre, Elkies, ..., so it's not clear (to me) why theoretical advances should also give linear growth!