When gasoline gets to $10.00/gallon, will the ⁹⁄₁₀ cent at the end go away?

No. That’s how and where the gubmint gits its mitts into us. 

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We asked the same question  when it was going to hit $1.00/gallon and the answer was no. I have no idea if anyone believes it makes the price look cheaper.

Probably not. The British equivalent is 0.1 penny, e.g. £1 78.9 a litre, and this has been used for years.

It is an old trading trick, going back many decades at least, so it's not going to end in a hurry.

For large prices, such as for furniture, the apparent reduction would be from a hundred to ninety-nine, Pounds or Dollars. For lesser values, pre-decimal shopkeepers in the UK often reduced the shillings by one penny: £10 19s 11d, rather than £11, for example.

So I don't think we are going to see the 1/10 cent or 1/10 penny disappear from filling-stations even as the numbers preceding it increase.

UK petrol prices are creeping towards £2 / litre (= £9.08 / UK gallon, which is bigger than the US gallon); but so far at least the oil companies seem to be trying to avoid going over that big psychological barrier. The filling-station still makes no more than a few pence profit from each litre, though; and over the years many garages even in suburban areas have stopped selling fuel to concentrate on servicing and car sales, or have gone out of business altogether.

A difficulty we have here (UK) is that the garages buy the fuels a month or more in advance of delivery, so even if the wholesale price drops the retail price does not usually drop for another month, if at all. The Government did cut the high duty slightly but it does not seem to have been passed onto the filling-stations.