EHB
For example, a game that uses EHB (Extra HalfBrite) mode during the game action (i. e. ingame) is Pinball Dreams. In my example run of the Beat Box table I can find the following in the copper list:
$00D6EA WAIT $4211FFFE
$00D6EE MOVE #$6200,BPLCON0
I. e. it waits for raster line $42 (66) and uses BPLCON0 to set the number of bitplanes to 6 without the HAM bit set, which means EHB mode.
Cannon Fodder uses EHB for some of its still pictures. I'm not sure if it uses it ingame, but I consider it unlikely.
A newer game, Nemac IV, presents you with a screen mode selection at the beginning, offering EHB mode as an option. It can then render its ingame 3D graphics to an EHB mode display. However, it requires a machine more powerful than a stock Amiga 500.
HAM
The way HAM (Hold And Modify) mode works makes it unsuitable for ingame animation in most cases. I used to be unaware of any game that uses it for gameplay. (Update: There were at least 2 games using HAM for their animated action. See robc's answer.)
Fury of the Furries uses HAM for its still images, for example the title image and the images that are displayed before entering a new world (each consisting of a set of levels).
The reason why it is unfeasible to use HAM for animated action is that you will either have to tolerate so-called color bleeding to the right of pixels or you will need to employ sophisticated algorithms to counteract the color bleeding, which timing constraints may forbid or which are just not worth it to come up with.
In HAM mode you usually use 6 bitplanes (you can also use it with 5 bitplanes, but there's not much point to that). That's 64 possible values per pixel. Values 0 to 15 select a full RGB color from the 16 color palette, the same way other normal screen modes would. However, value sets 16 to 31 (modify blue component), 32 to 47 (modify red component) and 48 to 63 (modify green component), respectively, each just change one of the color components relative to the color of the previous pixel. The other two color components remain unchanged from the previous pixel.
Say you set the color to palette entry 8 in column 100. Say also that this palette entry is $0333, i. e. a very dark gray. So the pixel in column 100 will appear gray.
Then, lacking a matching color in the palette, you modify the red component in column 101 to $6 (using a pixel value of $26 = 38). The resulting pixel in column 101 will now have a color equivalent to $0633, which is a dark unsaturated red.
Next, again lacking a matching color in the palette, you modify the green component in column 102 to $7 (using a pixel value of $37 = 55). The resulting pixel in column 102 will now have a color equivalent to $0673, which is a brown-greenish color.
Finally, lacking a matching color in the palette once more, you modify the blue component in column 103 to $6 (using a pixel value of $16 = 22). The resulting pixel in column 103 will now have a color equivalent to $0676, which is a dark gray with a very slight shimmer of green.

note that each differently colored square represents 1 pixel
So far so good, but say you now want to blit an animated character onto the bitmap. Say the rightmost pixel of that character ends up in column 100 and it is a bright white, palette value $0FFF. What this means is that the white will bleed into the next 2 pixels:
Due to modifying just the red component, the visible color in column 101 will be $06FF, a bright cyan.
Due to modifying just the green component, the visible color in column 102 will be $067F, a bright blue.
Only in column 103 will you have modified all three color components and the apparent pixel color will be the same $0676 as the original one before blitting.

note that each differently colored square represents 1 pixel
Depending on how many and which color components you consecutively modify, the color bleeding can affect more or less pixels and depending on how different the colors are to the originally intended colors, the artifacts can be more or less visible.
note that I scaled the image by a factor of 2 to make the artifacts easier to see
To counteract the bleeding, you would have to design a routine that figures out which pixels to the right of a blit to adjust and which pixel values are the optimal ones that let the apparent pixel colors match the original ones as closely as possible. You won't get completely rid of the artifacts, but they might now be unnoticeable unless you take a closer look.