Can the standard deviation be greater than the mean?

In a perfect normal distribution it can be.

In the ideal normal distribution ALL values are theoretically possible, from #-oo# to #+oo#.
And then any standard deviation #sigma# is possible

In the real world we work with datasets, that can often be well descibed by a normal distribution.

Say you have a filling machine for kilo-bags of sugar. The actual weight of the bags can be described as a normal distribution with a mean #mu# of 1000 gram.
In this case a #sigma# of more than 1000 would be unthinkable, as this would mean that 18% of your bags would have a negative weight!
(a machine this unreliable would be unthinkable anyway!).

Final answer:
In theory: YES -- in practice: (almost) NEVER

BTW :
In a standardised normal distribution the mean #mu# is calculated back to #0# and the standard deviation #sigma# is reduced to #1#, so there the answer is always YES.

Standard deviation is a measure of how "spread out" a distribution (or a data set) is. The mean is just where that distribution (or data set) is centered.

As a simple analogy, if we find out that 95% of newborn babies weigh 7.75 pounds, give or take 2.25 pounds, the value 7.75 is like our mean, and the 2.25 pounds is like the standard deviation—in fact, in this case it is twice the standard deviation (because two standard deviations left-and-right of a mean will be 95% of a Normal distribution).

So, all we need to do is imagine a data set where the average value is low, but the "elbow-room" is high. Something like, the daily high temperature (in °C) in winter in the US. The mean will certainly be low (near 0°C), but because the US covers a lot of latitude, the standard deviation will be high—some cities will see relatively warm winters, while others will have quite cold ones.