I suppose it is almost the definition of entrepreneurship that profit is sought and loss avoided. As soon as time is considered, this transforms into maximizing the value of the firm--accepting losses now in exchange for anticipated profits in the future. But the very logic of value maximization separates the individual entrepreneur from the firm. Sacrificing profits now for future profits only makes sense if it is possible to sell the firm, or more plausibly, sell an interest in future profits.
However, there is no presumption that a specific individual will save any part, much less all, of his or her income so as to maximize the increase in wealth. Yet the notion that such a motivation is central to "capitalism" is common. Weber's Protestant ethic explained economic growth in northern Europe due to the supposed Calvinist view that earning money was a sign of God's favor but spending it on luxuries reflected a sinful worldliness. This was in contrast to the supposed Catholic view that the whole point of earning money was to use it for something. (I think this "Catholic" view is more sensible and theologically sound, as well as being more consistent with modern economic reasoning.)
Even before, Marx more or less treated the income from profit as the same as saving and forecast eventual economic disaster because the growth of wealth will outstrip the investment, really industrialization, entailed by the inevitable development of the material productive forces. Hegelian hocus pocus really doesn't fit in well with modern economic thought.
Still, it is certainly possible that entrepreneurs would save a substantial portion of their profit. It seems that many entrepreneurs find their work enjoyable. They are really too busy to enjoy anything like the consumption they could afford. To the degree that effort and commitment are more likely to lead to success than a more dilatory approach, many of the most notable entrepreneurs will in fact have that characteristic. "Why waste my time on lavish consumption activity when I can plow money back into business activity have some real fun!" Some tycoons, including the President of the United States, have claimed that money is just a way of keeping score. On this view, the goal is to raise one's standing on the Fortune 500 list--perhaps to reach the top and win the game of life by earning the greatest net worth.
Since earning profit and avoiding loss generates value, value that is typically much greater than what the entrepreneur earns, it would seem that the rest of society is getting something for nothing. Fundamentally, if someone consumes income then resources must be devoted to produce the goods that are to be consumed. If the income is never consumed, then the resources can continue to be used for other purposes.
However, this scenario involves the entrepreneur saving all of this additional income, so it isn't available for anyone's consumption in the present. Presumably, it will be available for someone's consumption when the entrepreneur dies. So, if an entrepreneur doesn't consume profit but reinvests for fun, and these investments remain profitable, consumption is moved to the future. Leaving aside inheritance tax, whose future consumption can be increased is determined by the entrepreneur's bequest.
If the relevant comparison is to luxurious consumption by the entrepreneur now, it is difficult to see how this result is any worse. Of course, some might argue that some of these profits should be taxed and the proceeds transferred so that others might consumer more now.
Suppose, however, that that entrepreneur does not use these funds for direct investment, but simply saves them. There is no longer some story about an entrepreneur using the funds to enjoy business activity, but still, net worth increases as would more passive investment income. If financial markets function properly, any funds saved result in matching investment by some firm or other. The greater future output generated by that investment allows for additional consumption for whomever the entrepreneur leaves a bequest.
Unlike the situation where the entrepreneur is directly investing saved profits, these other avenues for saving leave the possibility of some kind of breakdown in financial markets such that saving and investment do not match with full employment of resources. "The" market interest rate might fail to fall enough to match "the" natural interest rate that coordinates saving and investment.
This sort of problem is always monetary broadly understood. Saving might occur by accumulating money, and if prices and/or wages are sticky, real output and employment will decline. In my view, an appropriate monetary regime avoids this perverse result. And so, the entrepreneur who has no interest in consumption but solely wants to save profit to "win" by accumulating the most wealth only causes problems if there is a less than ideal monetary regime.
That our driven entrepreneur allows for added consumption in the future for whomever he makes his bequest points to an alternative motivation. The entrepreneur might reduce consumption from what would be possible now with the intention of allowing for added future consumption for his descendants. But rather than simply allowing his children to consume, I would like to consider another motivation. The notion that his descendants will permanently earn sufficient capital income to maintain a high level of consumption forever.
While this seems reasonable enough on its face, I think this goal is better motivated by cultural factors that existed prior the modern market system. As the market order came to dominate, the aristocrats inherited a system that had evolved such that they earned rental income from land. There was no requirement that aristocrats do much to manage their land, much less become actively involved in business activities. Aside from living a life of leisure while being doted upon by many personal servants, the only real duty was to participate in what can be broadly understood as governance. The initial rationale of this system involved military service, and a norm of service as officers in modern armies was really a holdover.
While the time when this income must be rents from land is long past, the notion that one's children and grandchildren and so on can be set up to live in that fashion is a plausible motivation for saving. So, the entrepreneur earns profits and consumes little immediately, mostly saving. The point is to leave a sufficient bequest to provide for an adequate (high) income for ones children, They will then pass on a similar bequest to their children, and so on.
This motivation, however, does not result in capital income being saved in perpetuity. Once set up properly, the family must avoid dipping into principal (or capital,) but otherwise, the income is spent on consumption that may be lavish by the standards of many people but typical of the upper class.
The entrepreneur consumes less than he otherwise could and saves. The amount saved is never intended to be consumed, but rather is to form a principal that will generate an income that is intended to be consumed.
This motivation does not require that the entrepreneur enjoy making money or have any particular interest in direct investment. Still, any notion that this someone results in too much saving is still less plausible. The saving is transitional. Considering the "cultural" factors, the notion that the traditional aristocracy had more income than they could use is pretty implausible. Quite the contrary. Even so, if this transitional saving did outstrip investment, any problem with maintaining employment would still be a consequence of an inadequate monetary regime.
What would be the implication of a negative natural interest rate for entrepreneurs saving their profits? Assuming that the real market interest rate matches this natural interest rate, then the entrepreneur is making money to "keep" score, will provide a bequest to someone that allows for less real consumption than the entrepreneur could have enjoyed. Of course, if the entrepreneur continues with direct investment, he may well earn more than the natural interest rate. This is really only relevant with a more passive investment strategy. If the nominal interest rate is positive, but the real interest rate is negative due to inflation, then I suppose some entrepreneurs would count their nominal gains as making progress in the game. If the nominal interest rate is negative, however, simply holding onto wealth would be clearly counter-productive.
For the motivation of providing a bequest sufficient for one's descendants to earn a sufficient capital income, it would seem that a permanently negative natural interest rate would make it impossible. In such a scenario, refraining from consumption now provides no benefit and would earn no reward. Providing for one's children and more distant descendants would be possible, of course. But without a positive real interest rate, eventually the wealth will be gone. However, as is much more likely, the natural interest rate would only be transitionally negative and likely only for the shortest and safest assets.