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As David Hammen commented, the power emitted through Hawking radiation is proportional to $M^{-2}$. Thus the evaporation timescale for a black hole is proportional to $M^3$. This means that a more massive black hole is much more stable against evaporation than a lower mass black hole.

The other issue you mention is the limited rate that you can "feed" a black hole. There is inevitably a feedback; as gas is compressed towards the event horizon it gets hot and emits radiation. The pressure of this radiation can eventually balance the inward gravitational infall. For spherically symmetric accretion this leads to the Eddington limit, which sets the maximum spherical accretion rate, where $\dot{M}_{\rm max}\propto M$. That is, the maximum accretion rate is proportional to the black hole mass.

If accretion proceeds at the Eddington limit then the black hole mass grows exponentially with time and with a characteristic doubling timescale of around 50 million years (independent of the original mass - see this Physics SE page for some mathematical details).

If black holes were limited to this accretion rate (though there is some evidence from the presence of very luminous quasars at high redshift that they may exceed it), then the maximum black hole mass will depend on the age of the universe and the size of the initial "seed" black holes. If we assume an initial mass of 100 solar masses, a doubling timescale of 50 million years and that the seed black holes formed 400 million years after the big bang (all plausible, but contestable), then there have been 266 doubling timescales since and the black hole could have grown by a factor of $10^{80}$ !

Clearly there are no black holes with anywhere near this mass in the observable universe - the largest appear to be of order $10^{10}$ solar masses. Their growth is limited by their food supply. Supermassive black holes are found at the centres of galaxies. There is a poorly understood relationship between the black hole mass and the mass of the bulge of the galaxy it is in. The ratio peaks at about 1 percent for the most massive bulges (e.g. see Hu 2009; McConnell & Ma 2013). Since the most massive massive elliptical galaxies are of order $10^{12}$ solar masses, this appears to set the maximum mass of a black hole in the present day universe.

The future is speculation. If the cosmic expansion rate continues to accelerate, then galaxy mergers will become increasingly uncommon and the opportunities for further black hole growth will be limited.