How to losslessly combine multiple AVI files into a single file?

Try mencoder or ffmpeg, both free, both good.

Mencoder

mencoder -oac copy -ovc copy -o output.avi input1.avi input2.avi

FFmpeg

From the FFmpeg Wiki article on how to concatenate (join, merge) media files:

Create a file "mylist.txt" with all the files you want to have concatenated in the following form ( Lines starting with a dash are ignored ) :

# this is a comment
file '/path/to/file1'
file '/path/to/file2'
file '/path/to/file3'

Note that these can be either relative or absolute paths. Then you can encode your files with:

ffmpeg -f concat -i mylist.txt -c copy output

I use Avidemux for tasks like this.

Process with Avidemux is as follows:

1. Choose File->Open and select the first file

2. Choose File->Append and select the next file. Repeat this step until all the files you want to append to the first file have been appended.

3. Choose Save and select location and filename