Would this result send us into an endless feedback loop? Not really, because the tax is only a percentage of one's taxable income, and so the additional amount of tax triggered by any receipt will be smaller than the receipt itself. The additional amounts become smaller and smaller and eventually drop to just tiny fractions of a penny; at that point we can stop counting them.
For example, assume that the taxpayer is in the 35% marginal bracket, and she receives $10,000 of gross income (with no offsetting deductions) in a transaction in which the other party agrees to pick up the tax on the $10,000 receipt. That tax would be $3,500, and if the payor reimbursed the taxpayer for that amount, the additional $3,500 of income would trigger another $1,225 of tax -- 35% of $3,500. The tax on that $1,225 would be $428.75, the tax on that $428.75 would be $150.06, etc.
In the end, the total payout needed to allow the taxpayer to keep $10,000 after tax would be $15,384.62, calculated as follows:
Another way of looking at it is this: If this taxpayer gets $15,384.62 of additional gross income, the tax at 35% is $5,384.62 -- leaving the taxpayer with the $10,000 base payment to keep.
And so if the Clarks' receipt had been gross income, the attorney would have had to pay them more -- probably a substantial amount more -- to make them whole. That wasn't the result in Clark, but it would be the result if, say, one's employer agreed to pay one's income taxes. (See Reg. sec. 1.61-14.) If your employer agreed to pay $10,000 of income tax for you, then assuming a 35% tax, she'd have to pay you an additional $5,384.62 to pay the tax on the tax, the tax on the tax on the tax, and so on.
You math majors out there are way ahead of me -- you've already figured out the formula for finding the total payout (or PT). It is:
PT = PB + (.35 * PT)
where PB is the base payment (the amount that the taxpayer would like to keep). If PB is $10,000, the algebra goes like this:
PT = $10,000 + (.35 * PT)
.65 * PT = $10,000
PT = $10,000 / .65
PT = $15,384.62