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Funciton, 1222 bytes

Apart from numeric literals, there are two ways I can produce a value (any value at all) in Funciton: stdin and lambda expressions. Stdin is a single box while a full lambda expression requires more syntax, so I’m going with stdin. However, while stdin could be anything, all of the following work regardless of what input is provided.

All of the library functions used here existed before the challenge was posted.

0 (40 bytes in UTF-16)

╔╗┌┐
║╟┤└┼┬┐
╚╝└─┘└┘


This uses the raw syntax for less-than. A value is never less than itself, so the result of this is 0.

1 (52 bytes in UTF-16)

╔╗┌─╖┌─╖
║╟┤⌑╟┤ɕ╟
╚╝╘═╝╘═╝


⌑ returns a lazy sequence containing a single element and ɕ counts the number of elements. (The lazy sequence is lazy enough that this snippet doesn’t actually evaluate stdin at all!)

2 (70 bytes in UTF-16)

╔╗┌─╖┌─╖┌─╖
║╟┤⌑╟┤ʂ╟┤ɕ╟
╚╝╘═╝╘═╝╘═╝


= 2¹. ʂ generates all subsequences of a sequence, and thus turns a sequence of n elements into one with 2ⁿ.

3 (88 bytes in UTF-16)

╔╗┌─╖┌─╖┌─╖┌─╖
║╟┤⌑╟┤ʂ╟┤ɕ╟┤♯╟
╚╝╘═╝╘═╝╘═╝╘═╝


= 2 + 1. ♯ increments a value by 1.

4 (88 bytes in UTF-16)

╔╗┌─╖┌─╖┌─╖┌─╖
║╟┤⌑╟┤ʂ╟┤ʂ╟┤ɕ╟
╚╝╘═╝╘═╝╘═╝╘═╝


= 2².

5 (106 bytes in UTF-16)

╔╗┌─╖┌─╖┌─╖┌─╖┌─╖
║╟┤⌑╟┤ʂ╟┤ʂ╟┤ɕ╟┤♯╟
╚╝╘═╝╘═╝╘═╝╘═╝╘═╝


= 4 + 1.

6 (106 bytes in UTF-16)

╔╗┌─╖┌─╖┌─╖┌─╖┌─╖
║╟┤⌑╟┤ʂ╟┤ɕ╟┤♯╟┤!╟
╚╝╘═╝╘═╝╘═╝╘═╝╘═╝


= 3 factorial.

7 (110 bytes in UTF-16)

┌───┐┌─╖┌─╖┌─╖╔╗
│┌─╖├┤ɕ╟┤ʂ╟┤⌑╟╢║
└┤A╟┘╘═╝╘═╝╘═╝╚╝
 ╘╤╝


= A(2, 2) (Ackermann function).

8 (118 bytes in UTF-16)

┌────┐┌─╖┌─╖┌─╖╔╗
│┌──╖├┤ɕ╟┤ʂ╟┤⌑╟╢║
└┤&lt;&lt;╟┘╘═╝╘═╝╘═╝╚╝
 ╘╤═╝


= 2 &lt;&lt; 2 (shift-left).

9 (128 bytes in UTF-16)

┌───┐┌─╖┌─╖┌─╖┌─╖╔╗
│┌─╖├┤♯╟┤ɕ╟┤ʂ╟┤⌑╟╢║
└┤×╟┘╘═╝╘═╝╘═╝╘═╝╚╝
 ╘╤╝


= 3 × 3.

10 (146 bytes in UTF-16)

┌───┐┌─╖┌─╖┌─╖┌─╖┌─╖╔╗
│┌─╖├┤♯╟┤ɕ╟┤ʂ╟┤ʂ╟┤⌑╟╢║
└┤+╟┘╘═╝╘═╝╘═╝╘═╝╘═╝╚╝
 ╘╤╝


= 5 + 5.

42 (170 bytes in UTF-16)

┌──────┐┌─╖┌─╖┌─╖┌─╖┌─╖╔╗
│┌─╖┌─╖├┤!╟┤♯╟┤ɕ╟┤ʂ╟┤⌑╟╢║
└┤♯╟┤×╟┘╘═╝╘═╝╘═╝╘═╝╘═╝╚╝
 ╘═╝╘╤╝


= 6 × (6 + 1).