Attachment Rivit for rear derailleur
 

If the derailleur has play side to side and there is no way to tighten the rivet and the high limit has no more adjustment and you still can't get to the highest gear should the derailleur be replaced.

Which rivet, exactly, are you referring to?

Getting to the highest gear should never be a problem since the default state of the vast majority of RD's is for the spring to pull it to the highest gear. Perhaps you actually mean the lowest gear?

As mentioned above, what rivet?

...and are you sure this rivet is the source of play? There are other possible sources of slop, in the derailleur itself and how it mounts to your frame.

The high gear is the outermost (smallest) cog. It can't reach the closest cog to the derailleur? The low limit screw would affect whether or not the derailleur can reach the biggest cog (lowest gear).

If you're using an old derailleur, like a 5- or 6-speed unit, it may not be able to span newer, wider cogsets with more gears.

...and if your derailleur hanger's out of alignment, that can also affect whether or not the derailleur can reach all the cogs in the cluster.

There is a metal arm on the bicycle frame to which the derailleur is solidly bolted. The next position down on the derailleur has a rivet. One end is exposed the other end is under a plastic cover. The loose rivet allows the derailleur to rock side to side. That rocking prevents the derailleur from being adjusted to the (H) gear, the smallest one toward the outside.

The spring tension is pulling the derailleur toward the bicycle away from the smallest gear.

Not seeing this. Picture?

Bike and derailler brand?

This info will help get you an answer.

You're really going to have to post a labeled photo or video of what you're referring to. The "metal arm" is presumably the derailleur hanger. I suspect what you're calling a rivet is actually a pivot bolt.