I think we could fill pages with examples. How about Lorenz' discovery of chaos from the output of an interrupted computer run? Copying at length from Wikipedia:

Chaos Theory
In 1961, Lorenz was using a simple digital computer, a Royal McBee LGP-30, to simulate weather patterns by modeling 12 variables, representing things like temperature and wind speed. He wanted to see a sequence of data again, and to save time he started the simulation in the middle of its course. He did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To his surprise, the weather that the machine began to predict was completely different from the previous calculation. The culprit: a rounded decimal number on the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 printed as 0.506. This difference is tiny, and the consensus at the time would have been that it should have no practical effect. However, Lorenz discovered that small changes in initial conditions produced large changes in long-term outcome.

Lorenz's discovery, which gave its name to Lorenz attractors, showed that even detailed atmospheric modelling cannot, in general, make precise long-term weather predictions. His work on the topic culminated in the publication of his 1963 paper "Deterministic Nonperiodic Flow" in Journal of the Atmospheric Sciences, and with it, the foundation of chaos theory. He states in that paper:

Two states differing by imperceptible amounts may eventually evolve into two considerably different states ... If, then, there is any error whatever in observing the present state—and in any real system such errors seem inevitable—an acceptable prediction of an instantaneous state in the distant future may well be impossible....In view of the inevitable inaccuracy and incompleteness of weather observations, precise very-long-range forecasting would seem to be nonexistent.

His description of the butterfly effect, the idea that small changes can have large consequences, followed in 1969.

Lorenz's insights on deterministic chaos resonated widely starting in the 1970s and 80s, when it spurred new fields of study in virtually every branch of science, from biology to geology to physics. In meteorology, it led to the conclusion that it may be fundamentally impossible to predict weather beyond two or three weeks with a reasonable degree of accuracy. However, the recognition of chaos has led to improvements in weather forecasting, as now forecasters recognize that measurements are imperfect and thus run many simulations starting from slightly different conditions, called ensemble forecasting.

Of the seminal significance of Lorenz's work, Kerry Emanuel, a prominent meteorologist and climate scientist at MIT, has stated:

By showing that certain deterministic systems have formal predictability limits, Ed put the last nail in the coffin of the Cartesian universe and fomented what some have called the third scientific revolution of the 20th century, following on the heels of relativity and quantum physics.

Late in his career, Lorenz began to be recognized with international accolades for the importance of his work on deterministic chaos. In 1983, along with colleague Henry Stommel, he was awarded the Crafoord Prize from the Swedish Academy of Sciences, considered to be nearly equal to a Nobel Prize. He was also awarded the Kyoto Prize for basic sciences in the field of earth and planetary sciences in 1991, the Buys Ballot Award in 2004, and the Tomassoni Award in 2008. In 2018, a short documentary was made about Lorenz's immense scientific legacy on everything from how we predict weather to our understanding of the universe.

Proceedings of the National Academy of Sciences113, no. 31 (2016): E4446-E4454. $\endgroup$ – Joseph O'Rourke Dec 3 '19 at 0:501more comment