The standard normal ranges from $-\infty$ to $\infty$.

Your problem appears to be that your table doesn't go further.

Your question should therefore be modified to ask "How do I deal with the fact that my table doesn't go as high as my $Z$ value?

[Note that in your last paragraph, you have become confused. The region you're evaluating probability for is $Z<3.75$ but the boundary value of $Z$ you're trying to look up in the table (the $3.75$) is $>3$, as in your title.]

It seems like not having the value in your table would be a problem, but it's a very small one $-$ since your answer for $P(0<Z<3.75)$ can't be smaller than $P(0<Z<3)\approx 0.4999$ and can't be larger than $P(0<Z<\infty)=0.5$, you shouldn't have much difficulty narrowing the answer down to 3 significant figures of accuracy even so.

Additional accuracy (though I really don't think you need it) can be obtained by many methods. Here are three:

i) finding better tables (these seem to be of the same form as the ones you're apparently using)

ii) using a package that will evaluate standard normal cdfs for you. I just used R (simply typing pnorm(3.75) to obtain $P(-\infty<Z<3.75)$).

iii) using numerical integration to approximate the area between 3 and 3.75. For example, via Simpson's rule, a single interval (3 points) gives 0.0017 (the correct answer is 0.0013). Alternatively, because the density is convex, the integral will be bounded below by the midpoint rule and above by the trapezoidal rule, which usefully bounds where the answer can lie

But, really, just using the limits provided by 3 and $\infty$ is plenty, I imagine.

The standard normal ranges from $-\infty$ to $\infty$.

Your problem appears to be that your table doesn't go further.

Your question should therefore be modified to ask "How do I deal with the fact that my table doesn't go as high as my $Z$ value?"

[Note that in your last paragraph, you have become confused. The region you're evaluating probability for is $Z<3.75$ but the boundary value of $Z$ you're trying to look up in the table (the $3.75$) is $>3$, as in your title.]

It seems like not having the value in your table would be a problem, but it's a very small one $-$ since your answer for $P(0<Z<3.75)$ can't be smaller than $P(0<Z<3)\approx 0.4999$ and can't be larger than $P(0<Z<\infty)=0.5$, you shouldn't have much difficulty narrowing the answer down to 3 significant figures of accuracy even so.

Additional accuracy (though I really don't think you need it) can be obtained by many methods. Here are three:

i) finding better tables (these seem to be of the same form as the ones you're apparently using)

ii) using a package that will evaluate standard normal cdfs for you. I just used R (simply typing pnorm(3.75) to obtain $P(-\infty<Z<3.75)$).

iii) using numerical integration to approximate the area between 3 and 3.75. For example, via Simpson's rule, a single interval (3 points) gives 0.0017 (the correct answer is 0.0013 to 4dp). Alternatively, because the density is convex in this region (indeed, as whuber points out in comments, convex for $Z>1$ and $Z< -1$), the integral will be bounded below by the midpoint rule and above by the trapezoidal rule, which usefully bounds where the answer can lie

But, really, just using the limits provided by 3 and $\infty$ is plenty, I imagine.