Recently, the following question was asked: Without calculator, find out what is larger: $60^\frac{1}{3}$ or $2+7^\frac{1}{3}$. (Apologies; I don't know how to link to that question, but it is not essential for the question I am asking.)
Most people would not be able to extract cube roots without a calculator, unless the numbers were particularly easy, such as $64^\frac{1}{3}$ or $2+8^\frac{1}{3}$. But not using a calculator does not rule out doing some calculation.
As it turns out, the numbers in this case lend themselves to reasonably calculable approximations, which many people could perform in their heads, but might prove daunting to less experienced individuals.
So my question is, are the calculations I made reasonably within the intention of the restriction "without calculator?"
Please consider the difficulty of the calculations, not the total amount of calculation performed. Here is what I did, emphasizing the arithmetic calculation aspects: I cubed both quantities, leaving me to compare $60$ with $8+12(7^\frac{1}{3})+6(7^\frac{2}{3})+7$.
Collecting terms and rearranging, the original question becomes one about a quadratic equation:
Is $$x^2+2x-7.5$$ greater or less than $0$ when $x=7^\frac{1}{3}$?
This in turn becomes: Is $7^\frac{1}{3}$ greater or less than $r$, the positive root of $$x^2+2x-7.5=0$$.
By the quadratic formula $$r=(-2+\sqrt{4+30})/2$$.
Although the square root of $34$ may look like one of those calculations that would require a calculator, it turns out that determining a precise value would just make subsequent calculations dependent on a calculator as well.
By good fortune (or the cleverness of the original poser of the question), $34$ is close to $36$, so we may approximate $\sqrt{34}$ as $(6-a)$.
Thus we look for $$34=36-12a+a^2$$.
But since $a$ will be small compared to $6$, we can approximate by ignoring the $a^2$ term and calculate $a=\frac{1}{6}$. it is easy to see that $(6-\frac{1}{6})^2$ exceeds $34$ by $\frac{1}{36}$. Again, by seeming good fortune, the next reasonable fraction greater than $\frac{1}{6}$ is $\frac{17}{100}$.
$$(6-\frac{17}{100})^2$$ is also calculable as $$36-\frac{204}{100}+\frac{289}{10000}$$. Since the second term decrements $36$ by $2.04$ and the third term only restores $0.0289$, we see that $(6-\frac{17}{100})^2$$ is less than $34$. So $$(6-\frac{1}{6})>\sqrt{34}>(6-\frac{17}{100})$$, hence $$(2-\frac{1}{12})>r>(2-\frac{17}{200})$$.
What remains is to cube the numerical values bracketing $r$ and compare the results to $7$.
$$(2-\frac{1}{12})^3=8-1+\frac{1}{24}-\frac{1}{1728}$$ which is greater than $7$ by observation.
$$(2-\frac{17}{200})^3=8-\frac{204}{200}+6(\frac{289}{40000})-(\frac{17}{200})^3=8-\frac{204}{200}+(\frac{1734}{40000})-(\frac{17}{200})^3$$.
The arithmetic is a little harder here, but the first and second terms are less than $7$ by $0.02$ and the third term is reasonably seen to be greater than $0.04$, making the sum of the first three terms greater than $7$ by at least $0.02$. The last term is certainly smaller than $(\frac{20}{200})^3$ which is $0.001$, so the sum of the terms is greater than 7.
This means that $r^3>7$ or $$r>7^\frac{1}{3}$$. From this, the original question can be answered. In performing calculations, no roots were extracted, but binomial expressions up to cubes involving fractions were calculated. I personally found the numbers in the numerators and denominators tractable, but would this be considered by the community as being in the spirit of "without calculator?"
There is a huge difference between mental calculation and calculation with pencil and paper. The storage available with paper is huge. Taking one of your examples, if I were asked to calculate $\sqrt{34}$ mentally (and I do a lot of mental calculation) I would do $\sqrt {34}=6\sqrt{1-\frac{2}{36}}\approx6(1-\frac1{36})=6-\frac 16 \approx 5.83$ and I couldn't do much better. With pencil and paper one can do the old square root digit-by-digit algorithm or Newton's method and get as many decimals as you wish rather quickly.
It helps to have a bunch of constants at the tip of your brain. Here is my list. As I said there, they came from experience, not from sitting down to memorize them.
It depends a lot on how much time you have for a problem. If you are doing a Putnam problem where you have 30 minutes you can do quite a bit of computation if that seems appropriate. If you are doing an exam with $1$ minute per problem, not so much.
Often there is something you are expected to notice to make it easier. Your example of which is larger, $60^{1/3}$ or $2+7^{1/3}$ begs you to cube them and compare the results. It sure looks like using $7^{1/3} \lt 8^{1/3}=2$ will get you home without computing anything, but that is not good enough. $$\left(2+7^{1/3}\right)^3=8+12\cdot 7^{1/3}+6\cdot 7^{2/3}+7$$ which is less than $63$. We need a better value for $7^{1/3}$. I don't know the old method for cube roots by hand, so I would say $7^{1/3}=2(1-1/8)^{1/3}\gt 2(1-1/24)\approx 1.92$, plug that in and find the right side is greater than $60.1584$. I think this is the expected approach.