Question: Single person - T-test?

Can you look at improvement in scores across two time periods, if you have hundred of data points but they are all from a single individual. Can I do a t-test to compare the indvidual's scores over one period to the same individual's scores over a later period?
I've never encountered anything like this before (a single person making up all the data).
If you have lots of measurements of the length of a child at age 9 and lots of measurements of the same child at age of 10, then, you can evaluate if this child has grown by doing a t-test. So, yes you can.

(The question is, what is the unit of measurement? it is often a person, but here it is the length measurement. If you randomize an selects a few countries in the Americas and a few in Africa (and measure something) then the unit of investigation will be the country and you can test it with a t-test (provided that the data are normally distributed etc.)
I'm measuring "number of interactions with a device" (it's an exercise used with autistic children, and the more interactions they have with this device on a daily basis, the more they have improved in their skills). So basically, I have totals for every day for about 2 years, and I want to compare the average number of interactions with the device in the first year to the average number in the second year. Is a t-test a valid way to do this? Would it be a paired samples t-test given that it's the same person in the first and second year? Or, do I use a totally different test, given that this is a single person over time?
No it will not be a "paired sample". You have only one individual. The question is: has this individual improved?

It does not seem ok to just lump first year in one group and the second year in an other group. I would guess the the change ha been more gradual.
Why don't you plot the data with the time on the x-axis and the "number of interactions with a device" on the y-axis. Show us the plot. Maybe a linear regression model will fit?