While Loop Crashes RBX Studio?

Perhaps the reason this is happening is because you create new event connections every execution of your while loop and the wait occurs within that event connection.

Take a look at your loop. The way your code is structured you have.

while
 - do calculation
 - event connection
   - wait

The wait function will now only run when the event is run (and that's actually done in a seperate thread so it won't wait in the while function).

The way to correct this is to move your event connection outside the while loop and leave a wait() or wait(10) within the while loop.

local i = 1
while i == 1 do
    local percent = 4 -- Percentage to multiply by.
    local base = 800 -- Base value.
    local expTime = 1 -- Time in units to use as the exponent.
    local differ = base + (base * (percent / 100)) -- To change to exponential growth, change the - to +.
    local rate = differ / base
    local answer = base * rate ^ expTime
    local newAnswer = answer>=0 and math.floor(answer+0.5) or math.ceil(answer-0.5)

    wait(10) -- left this here
end

-- moved this block outside
game.Players.PlayerAdded:connect(function(player)

local leaderstats = Instance.new("IntValue")
leaderstats.Name = "leaderstats"
leaderstats.Value = 0


local cash = Instance.new("StringValue")
cash.Name = "cash"
cash.Value = newAnswer

leaderstats.Parent = player
cash.Parent = leaderstats
end)

EDIT You suggested that the script was not working the way you expected. Perhaps this is what you expected.

game.Players.PlayerAdded:connect(function(player)

local i = 1

local leaderstats = Instance.new("IntValue")
leaderstats.Name = "leaderstats"
leaderstats.Value = 800

local cash = Instance.new("StringValue")
cash.Name = "cash"

cash.Value = newAnswer

leaderstats.Parent = player
cash.Parent = leaderstats
local expTime = 0 -- Time in units to use as the exponent.
while i == 1 do
    expTime = expTime + 1
    local percent = 4 -- Percentage to multiply by.
    local base = 800 -- Base value. 
    local differ = base + (base * (percent / 100)) -- To change to exponential growth, change the - to +.
    local rate = differ / base
    local answer = base * rate ^ expTime
    local newAnswer = answer>=0 and math.floor(answer+0.5) or math.ceil(answer-0.5)
    leaderstats.Value = newanswer
    wait(1)
end

end)