On Tuesday I gave geometry the following opener.
I let them think about it for 5 minutes or so. After I asked them for smaller cases. We looked at 2^8 vs 8^2 since some were claiming whichever number had the larger base would have the larger value.
After a few more minutes of discussion they plugged them into their calculators and checked their claims as to which was larger. The issue was the answers were all in scientific notation, sure it says a lot about how much larger one is than other, but students were not able to see how larger the numbers actually were.
So I pulled up Wolfram Alpha.
What students found most interesting was the number name. I had no idea these existed… We tried to find where a google would be….
This turned out to be a fun opener!
Quadrant Dan 
Ideas for those who want follow-ups:
(1) are there any numbers (feel free to restrict to integers) where a < b but a^b < b^a? What are they?
(2) approximating big powers
A rough approximation that can be really helpful is 2^10 is close to 1000 (10^3). For 42^34, you could approximate:
42 is approximately 40 = 4 * 10
So 42 ^ 34 could be close to 2^ 68 * 10 ^34
Using our approximation of 1000 for 1024, replace 2^68 by 2^8 * 1000^6, so we get
256 * 10^52 or 2.56 * 10^54
Of course, that's still only about 1/6 the precise value calculated by worlfram alpha, but seems pretty good for such simple calculation.
(3) approximating compound interest
let's say you want to do better than the previous approximation (we do, we do!) Can we make a useful adjustment to correct for replacing 42 by 40? Well, 40 = 40 * 1.05, so 42^34 = 40^34 * 1.05^34.
That second term looks like a calculation for compound interest, right? one rule of thumb (the rule of 70) is that a compounding process will double in approximately (70/rate) periods. In other words, the time it takes your money to double at interest rate r% is about 70/r years. At 5%, about how many doubling periods do we get when we compound 34 times? About 34/70 * 5 which is about 2.5. So, we can approximate 1.05^34 by 2^2.5
Depending on your love for sqrt(2), you ignore that bit and end up with a final estimate of 10^55 (approximately 2.56*10^54 * 4). However, for those playing along who want to say sqrt(2) is close to 1.5, then we get a final approximation of 1.5*10^55.
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